Application of Complex Vectors and Complex Transformations in Solving Maxwell s Equations

Transcription

1 Applcaton of Complex Vectos and Complex Tansfomatons n Solvng Maxwell s Equatons by Payam Saleh-Anaa A thess pesented to the Unvesty of Wateloo n fulfllment of the thess equement fo the degee of Maste of Appled Scence n Electcal and Compute Engneeng Wateloo, Ontao, Canada, 00 Payam Saleh-Anaa 00

2 I heeby declae that I am the sole autho of ths thess. Ths s a tue copy of the thess, ncludng any equed fnal evsons, as accepted by my examnes. I undestand that my thess may be made electoncally avalable to the publc

3 Abstact Applcaton and mplcaton of usng complex vectos and complex tansfomatons n solutons of Maxwell s equatons s nvestgated. Complex vectos ae used n complex plane waves and help to epesent ths type of waves geometcally. It s shown that they ae also useful n epesentng nhomogeneous plane waves n plasma, sngle-negatve and double-negatve metamateals. In specfc I wll nvestgate the Otto confguaton and Ketschmann confguaton and I wll show that n ode to obseve the mnmum n eflecton coeffcent t s necessay fo the metal to be lossy. We wll compae ths to the case of plasmon-le esonance when a PEC peodc stuctue s llumnated by a plane wave. Complex tansfomatons ae cucal n devng Gaussan beam solutons of paaxal Helmholtz equaton fom sphecal wave soluton of Helmholtz equaton. Vecto Gaussan beams also wll be dscussed shotly.

4 Acnowledgements I would le to than my supevso Pofesso Safeddn Safav-Naen fo hs suppot and encouagement n the couse of ths thess. He was moe than just a supevso fo me; he gave me feedom to choose the topc of my nteest, and taught me nvaluable lessons about eseach and lfe. I would le also to than Pofesso Sujeet K. Chaudhu and Pofesso A. Hamed Majed fo evewng my thess and the nvaluable comments and the tme and effot. v

5 Table of Contents Lst of Fgues...v Lst of Tables...x Lst of Abbevatons...x Complex Vectos.... Revew of Complex Lnea Algeba.... Vecto Identtes fo Complex Vectos Paallel and Pependcula Vectos Polazaton of Complex Vectos Dot Poduct and Coss Poduct fo Complex Vectos... 6 Plane Waves wth Complex Wave Vectos...8. Intoducton Complex Plane Wave Soluton of Maxwell s Equatons Complex Poyntng Vecto LP Wave Vecto NLP Wave Vecto TE/TM Decomposton Geometcal Repesentaton of Complex Wave Vecto fo Dffeent Types of Meda... 3 Complex Plane Wave Soluton of Maxwell Equatons n Pesence of Plane Boundaes Complex Wave Vecto Matchng Ctea fo Desgnatng Waves as Incdent, Reflected and Tansmtted Geometcal Repesentaton of Complex Wave Vectos n Plane Bounday Poblem v

6 3.. Plane Bounday between Two Medum of Type I Plane Bounday between Medum of Type I and II Plane Bounday between Medum of Type I and III Poyntng vecto of supeposton of ncdent and eflected waves Supeposton of ncdent and eflected wave n TE y mode Supeposton of ncdent and eflected wave n TM y mode Supeposton of ncdent and eflected wave n mxed mode Plane waves n plasma and suface plasmon and the popetes Suface plasmonc esonance Otto and Ketschmann confguatons fo exctng suface plasmons Complex Tansfomatons and Gaussan Beams Devaton and Popetes of Scala Gaussan Beam Complex lnea tansfomatons Vecto Gaussan Beam Summay and Futue Reseach Summay Futue wo Refeences v

7 Lst of Fgues. Dffeent confguatons of complex wave vecto n type I medum.... Dffeent confguatons of complex wave vecto n type II medum Dffeent confguatons of complex wave vecto n type III medum Dffeent confguatons of complex wave vecto n type IV medum Two smple medum wth plane bounday between them Detemnng ncdent eflected and tansmtted waves usng powe flow cteon Two smple medum of type I wth plane bounday between them Incdent, eflected and tansmtted wave at nteface between a lossless delectc and lossless DNG metamateal (both meda ae type I) Two smple medum of type I and II wth plane bounday between them Plane bounday between a lossless delectc and a lossless SNG metamateal (Plasmons) Two smple medum of type I and III wth plane bounday between them Plane bounday between a lossless delectc and a hghly lossy delectc (Zennec Waves) The a gap between two delectcs wth wave vectos and aveage tme Poyntng vectos. The z component of Poyntng vectos n all thee meda s constant and equal Nomal ncdence on a sngle negatve metamateal slab. All Poyntng vectos ae constant and equal...45 v

8 3. A delectc slab n the a, wth wave vectos and total tme-aveage Poyntng vectos. In ths case thee s no flow of powe nomal to the boundaes Wave vectos, tme aveage Poyntng vecto and magnetc feld n patal eflecton fom nteface of two lossless delectcs Wave vectos, tme aveage Poyntng vecto and magnetc feld fo total eflecton, fom nteface of two lossless delectcs Suface Plasmon Polatons and Suface Plasmon Resonance Ketschmann s confguaton fo exctng suface plasmons Polatons wth lossless metal Magntude of total eflecton coeffcent as a functon of ncdent angle n Ketschmann s confguaton, when the metal s lossless Magntude of total eflecton coeffcent as a functon of ncdence angle n Ketschmann s confguaton, when the metal s lossy Wave vectos and Poyntng vectos n Ketschmann s confguaton when the metal s lossy Otto s confguaton fo exctng suface plasmon Polatons wth lossless metal Reflecton coeffcent vs. ncdent angle n Otto s confguaton, when the metal s lossless Reflecton coeffcent vs. ncdent angle n Otto s confguaton, when the metal s lossy Wave vectos and Poyntng vectos n Otto s confguaton when the metal s lossy...64 v

9 Lst of Tables. All possble polazatons fo paametes of a plane wave... x

10 Lst of Abbevatons LP: Lnea Polazaton- Lnealy Polazed NLP: Nonlnea Polazaton- Nonlnealy Polazed CP: Ccula Polazaton- Cculaly Polazed TE: Tansvese Electc TM: Tansvese Magnetc SNG: Sngle Negatve (Metamateal) DNG: Double Negatve (Metamateal) PHE: Paaxal Helmholtz Equaton GB: Gaussan Beam SAGB: Smple Astgmatc Gaussan Beam GAGB: Geneal Astgmatc Gaussan Beam x

11 Chapte Complex Vectos. Revew of Complex Lnea Algeba In ths thess vectos wth eal components ae called eal vectos and vectos wth complex components ae called complex vectos. Although phasos that ae complex vectos ae vastly used n tme-hamonc Electomagnetc, thee has not been enough consdeaton egadng dffeences between eal vectos and complex vectos. Thee ae two types of epesentaton fo vectos wth eal components: Component-wse epesentaton and geometcal epesentaton. It s possble to defne opeatons such as nne poduct, coss poduct, length etc. between vectos usng one epesentaton and deve t n anothe epesentaton. Fo example we can defne nne poduct of two vectos usng component epesentaton of vectos and deve ts geometcal fomulaton and vce-vesa. Lnea algeba technques ae used to deal wth the vectos component-wse. Fo example the geometcal opeatons such as otaton and tanslaton can be epesented by matxes wth eal components whch we wll call them eal matxes hencefowad. The stuaton s dffeent fo complex vectos because thee s no geometcal epesentaton fo a complex vecto unless we decompose t to ts eal pat and magnay pat that ae eal vectos.

12 To defne the opeatons such as nne poduct, coss poduct, length etc. between complex vectos we have to use the coespondng component-wse defnton fo eal vectos wth some modfcaton. Fo nne poduct we have to modfy the defnton by mang the components of the second vecto conjugates. V = v x ˆ W = w x ˆ VW= v w ( ) * * * VW = V W (.) V * = V V = v The defnton of nne poduct guaantees that the length s a non-negatve eal numbe. A complex vecto can be wtten n tems of ts eal and magnay pats: V = V + j V (.) n whch V and V ae eal vectos. * V = V V = V + V = v (.3) but j v VV = V V + VV= (.4) V V =0 * V V = j V V (.5) Fom (.4) we obseve that t s possble to have can have V 0. VV = 0 wth

13 The best geometcal epesentaton that we can have fo a complex vecto s by epesentng ts eal pat ( V ) and magnay pat ( V ).. Vecto Identtes fo Complex Vectos Most of the denttes that ae vald fo eal vectos ae also vald fo complex vectos wth some exceptons. Fo example the mplcaton VV = 0 V= 0 s not vald fo complex vectos. Rathe ths statement s tue: VV * = 0 V= 0 A multlnea functon F of vecto aguments that s lnea n evey agument: V (=,,...,n) s a functon ( V, V,, αv + βv,, V ) = α ( V, V,, V,, V ) + β ( V, V,, V,, V ) F F F n n αβ, A multlnea dentty s of the fom (,,,,, ) F V V V V = 0 n (.6) Theoem: All multlnea denttes that ae vald fo eal vectos ae also vald fo complex vectos. Fo example the dentty: n A ( B C) = B( AC) C( A B) (.7) s also vald fo complex vectos. Havng AX and (A X) n whch A s a nown vecto and X s unnown, usng (.7) we have 3

14 X= ( ) ( ) AX A+ A X A AA whch s vald fo both eal and complex vectos, but snce wthout vectos: (.8) AA can be zeo A beng zeo we deve an altenatve fomulaton fo complex X= ( ) ( ) A X A+ A X A * * AA * (.9) Equaton (.8) mples that A X= 0 and A X=0 A=0 o X=0 * Equaton (.9) mples that A X= 0 and A X=0 A=0 o X=0.3 Paallel and Pependcula Vectos We now that fo paallel eal vectos vectos AB=0 A B=0 and fo pependcula eal. The same defntons ae used fo complex vectos but obvously the geometcal ntepetaton doesn t exst anymoe. A B=0 and A 0 α, B= αa (.0) 3 AB= 0 and A 0 C, B=C A (.) We can calculate α and (.7). C n equatons (.0) and (.) usng equaton ( ) ( ) ( ), A B=0 α A A B = A A B -B A A * * * * * * ( ) ( ) ( ) = = * A B A A B = A A B -B A A C AA * * A B AA * (.3) (.) 4

15 .4 Polazaton of Complex Vectos It s vey mpotant to notce that the defnton gven hee fo the tem polazaton s dffeent fom the classcal defnton of polazaton whch assumes the complex vecto s a phaso that epesents a eal vecto feld wth hamonc tme dependency. Howeve ths geneal defnton concdes wth the standad defnton of polazaton of a phaso when the complex vecto s a phaso. Defnton: A complex vecto gven as V = V + j V only f V V=0, whch means V and V ae paallel. s lnealy polazed (LP) f and It can be shown that a complex vecto s LP f and only f t can be wtten as multplcaton of a complex scala and a eal vecto. V= ca, c, A 3 (.4) Defnton: A complex vecto gven as = + j V V V s nonlnealy polazed (NLP) f V V 0, whch means V and V ae not paallel and fom a plane. We defne a cuve usng paamete ϕ [0, π ], and eal vecto v ( ϕ ) as ( ϕ ) = Re( e jϕ ) = cos( ϕ) sn ( ) v V V V ϕ (.5) Ths cuve s n the plane fomed by V and V (polazaton plane) and to fnd equaton of ths cuve we defne to auxlay vectos a and also n polazaton plane: (.6) b whch ae 5

16 a= V ( V V ) b= V ( V V ) Wth some staghtfowad vecto opeatons we can fnd the equaton of the cuve as: ( ( ϕ) ) ( ( ϕ) ) 4 a v + b v = V V (.7) Equaton (-7) s equaton of an ellpse n the polazaton plane. If V same. If s multpled by a complex scala the polazaton ellpse wll eman the V V =00 polazed (CP) the cuve wll be a ccle and vecto V s called cculaly The followng popetes ae staghtfowad to pove: V s LP * = V V 0 V s CP V V = 0.5 Dot Poduct and Coss Poduct fo Complex Vectos Real pat and magnay pat of the complex vectos ae smla to eal and magnay pat of the complex vectos: Re ( ) * ( * A = A+A ) Im( ) = ( ) Α A-A (.8) j Thee s no coespondng quantty to absolute value and agument (phase) of complex numbes unless we consde the components. Conjugaton dstbutes ove the standad vecto opeatons: ( ) ( AB ) * * * A B = A B * * * =AB (.9) 6

17 3 3 Usng (.8) and (.9) t s easy to pove that f A and B then we have: Re( AB) = Re ( A) B, Im( AB) = Im ( A) B (.0) ( ) ( ) ( ) ( ) Re A B = Re A B, Im A B = Im A B (.) If V and W ae two LP vectos wth dffeent polazatons then we have: V+ jw= 0 V = 0 W= 0 (.) If V s LP and W= W + jw s NLP then we have: V W = 0 V W = 0 V W = 0 (.3) VW = 0 VW = 0 VW = 0 Havng eal unt vecto (.4) we can wte any gven complex vecto as summaton of LP vecto paallel to û and anothe vecto that s nomal to û. In (-8) we put A = û : û ( ) ( ) X= uˆ X uˆ+ uˆ X û (.5) The fst tem s a LP vecto paallel to û. û and the second tem s nomal to Let s defne C=A B If A and B ae LP then C s also LP. If A s LP but B s NLP then C s also NLP wth the same plane of polazaton as B and wth C pependcula to B and C pependcula to B. 7

18 Chapte Plane Waves wth Complex Wave Vectos. Intoducton In ths chapte I would le to use the denttes and popetes of the complex vectos that wee ntoduced n chapte. When total eflecton happens at the nteface between two delectcs the efacted wave would be an evanescent wave and usually s studed by defnng a fcttous angle that ts sne o cosne s geate than one. Usng complex vectos we show that popagatng plane waves and evanescent plane waves ae dffeent cases of a plane wave wth complex wave vecto. Usng the concept of complex wave vecto I wll povde a unfed appoach fo analyss of multlayeed meda consstng layes of delectc, meta- mateal (sngle-negatve and double negatve) and hghly conductve medum and actve medum. If a complex vecto feld has ths fom: ( ) f ( ) E E = 0 (-) n whch E s a constant complex vecto and f ( ) s a complex scala feld. 0 Then the polazaton popetes of complex vecto feld E ( ) ae detemned by E 0 and they don t change wth poston. Popetes of scala 8

19 feld f ( ) such as sufaces of constant phase and sufaces of constant ampltude ae also attbuted to the complex feld E. ( ) We must eep n mnd that all complex vecto felds cannot be wtten the fom of equaton (-) necessaly and so popetes such as sufaces of constant phase and ampltude ae not defned n geneal fo complex vecto felds.. Complex Plane Wave Soluton of Maxwell s Equatons We stat wth souce-fee phaso-doman (fequency-doman) Maxwell Equatons n a smple (lnea, homogenous, sotopc) medum wth complex pemttvty ε and pemeablty μ egadless of ealsablty of them n eal wold. The tme dependence ( exp j t) ω s assumed hencefowad. E= jωb H= jωd D = 0 B = 0 B= μh D = εe (-) and f we assume that the scala functon n equaton (-) has ths fom: f = exp( j (-3) ( ) ) n whch s a complex vecto. 9

20 Substtutng E ( ) = E0 exp( j ) n Maxwell equatons afte some staghtfowad smplfcaton we obtan: E= ωμh H = ωεe E = 0 H= 0 (-4) And wth moe smplfcaton we ave at: = ωμε= (-5) Equaton (-5) shows that the wave vecto only depends on the poduct με athe thanε and μ. The wave numbe s also defned as: ωμε =, whch n geneal s a complex numbe. It s vey mpotant to notce that με can be eal o pue magnay egadless of ε and μ beng complex, eal o pue magnay. By wtng = + j (-6) wth espect to ts eal and magnay components we have Fom (-5) and (-6) we obtan: = ω Re = ω Im ( με ) ( με) (-7) Equaton (-7) shows that t s useful to consde the specal cases that με s eal o pue magnay ( Re( με ) = 0 o Im( με ) = 0). Based on ths we can ecognze fou types of mateal: Type I Type II Type III Type IV ( με ) ( με ) Re > 0 Im = 0 ( με ) ( με ) Re < 0 Im = 0 ( με ) ( με ) Re = 0 Im < 0 ( με ) ( με ) Re = 0 Im > 0 0

21 In types I and II με s eal and n types III and IV με s pue magnay. Lossless delectcs and double negatve meta-mateals fall nto the type I. Plasma and sngle-negatve meta-mateals ae of type II. Hghly conductve (lossy) meda belong to type III. Hghly actve meda ae n type IV. But these categoes ae not lmted to above-mentoned examples nonetheless. Hee ae fou cases of ε and μ that esults n medum of type I: ε = ε ε = ε ε = jε ε = jε μ = μ μ = μ μ = jμ μ = jμ (ε, μ,ε and μ ae postve eal numbes) Anothe equvalent fom of expessng the condtons fo dffeent types of meda s as follows: ε = ε exp ( jδ ε ) μ = μ exp( jδ μ ) δε + δμ = mπ δε + δμ = ( m+ ) π π δε + δμ = mπ + 3π δε + δμ = mπ + typei typeii typeiii typeiv.3 Complex Poyntng Vecto In ths secton fst the most geneal fomulaton fo complex Poyntng vecto: ( ) = exp( j ) E E and = = exp( j ) 0 We obtan: H E H0 (-8) ωμ

22 * * S ( ) = * exp ( ) E0 ( E0 ) (-9) ωμ And usng dentty (.7) t can be wtten as: ( ) ( ) * ( * * exp * S = ) E ωμ 0 E0 E0 (-0) Smlaly we can obtan an altenatve fomulaton wth espect to magnetc felds: exp ωε S ( ) = ( ) H * ( H ) 0 0 (-) ( ) exp ( ) * S = ( ) H ωε 0 H0 H0 (-).3. LP Wave Vecto If s lnealy polazed, whch means and ae paallel: = 0 (-3) 0 * Snce s LP, E =0 esults n E = 0 egadless of polazaton of E 0. Smlaly we have H * =0. Theefoe equatons (-0) and (-) become: * S = exp * ( ) exp( ) ωμ = ωε E0 H0 (-4) Theefoe Assumng s lnealy polazed wth same polazaton as ε = ε jε and μ = μ jμ, the aveage of tme-doman S Poyntng vecto can be calculated as:

23 S ωε (, t) = Re S( ) = exp( ) H ( ε ε ) = exp ωμ ( ) E ( μ μ ) 0 0 (-5) We have the followng obsevatons: - When both and ae non-zeo, decton of flow of powe ( ) and decton of changng ts ampltude ( ) ae the same. In othe wods the planes of constant ampltude and planes of constant phase ae paallel. -Fo meda of type I and II we have = 0 equaton (-3) mples that =0 o = 0. whch togethe wth 3- Fo type I meda usng equaton (-7) only =0 s acceptable. Ths mples that thee s no change n ampltude. In othe wods lnealy polazed wave vecto n meda of type I s always a eal vecto. The complex Poyntng vecto usng equaton (-4) s: S= E = * 0 H0 (-6) ωμ ωε whch s a constant LP vecto n the decton of = The aveage tme-doman Poyntng vecto smplfes to the followng: ε μ S (, t) = H 0 = E 0 (-7) ωε ωμ Dependng on the sgn of ε, the tme aveage Poyntng vecto S (,t) and ae paallel o ant-paallel. 4- Fo type II based on equaton (-7) only =0 s acceptable. In othe wods lnealy polazed wave vecto fo type II meda s pue magnay. The complex Poyntng vecto usng equaton (-4) s: 3

24 j j S= exp * ( ) E0 = exp( ) H0 ωμ ωε (-8) The complex Poyntng vecto s a LP vecto wth same polazaton as. The aveage tme-doman Poyntng vecto becomes: ε S (, t ) = Re ( ) = exp ( ) S H0 ωε μ = exp ( ) E ωμ 0 (-9) So f ε = 0 thee wll be no flow of powe n meda of type II. 5- Fo type III and IV meda, both and ae non-zeo. They ae antpaallel fo type III and they ae paallel fo type IV. If s LP, E and H ae both LP (and nomal to each othe) o both ae 0 0 NLP wth the same plane of polazaton whch s nomal to..3. NLP Wave Vecto Both and ae nonzeo because othewse would be LP. If s NLP, E and H cannot be LP at the same tme, so we dstngush the 0 0 followng thee cases - E s LP: 0 * E = 0 E = 0 E = 0 E = (-0) 4

25 Ths mples that E0 s pependcula to plane of polazaton of. E0 s also nomal to plane of polazaton of. Theefoe H and have the same H0 0 plane of polazaton. Usng equaton (-0) we can smplfy the geneal equaton fo complex Poyntng vecto (equaton (-0)) as follows: * S ( ) = * exp ( ) E0 (-) ωμ and the aveage tme-doman Poyntng vecto s obtaned as: S (, t) = Re ( ) = exp ( ) ( μ μ ) S E0 (-) ωμ We obseve that S (,t) s a functon of poston and t s paallel to f μ = 0. In ths case fo meda of type I and II n whch = 0 the decton that S (,t) decays s and the decton of the flow of powe s. Snce the decton of S (,t) s paallel to the plane of constant (,t) ths type of wave s called suface wave, whch s the chaactestc of the wave n nfnte secton of delectc wavegude stuctues. S The ctea fo the popagaton to be attenuatng, lossless o amplfyng, s detemned by sgn of S (,t). If S (,t) s negatve the popagaton s lossy because S (,t) deceases n the decton of flow of powe. If t s postve the popagaton s amplfyng because S (,t) nceases n the decton of flow of enegy. If (,t) S (,t) s constant n the decton of popagaton. S s zeo the 5

26 S (, t) > 0 amplfyng popagaton S (, t) < 0 attenuatng popagaton S, = 0 lossless popagaton ( t) Usng equaton (-) sgn of (,t) S s detemned by ( ). μ μ = μ. μ So the ctea fo type of popagaton (amplfyng, attenuatng, and lossless) n geneal depend on both medum and wave popagaton vecto: ω μ με μ ( ) ω μ με μ ( ) ω μ με μ ( ) Im > 0 amplfyng popagaton Im < 0 attenuatng popagaton Im = 0 lossless popagaton Hence fo meda of types I and II that we have. = 0 the sgn s ndependent of wave vecto and only depends on medum. Medum s lossy f μ > 0 t s lossless f μ =0 and t s actve f μ < 0. - s LP: H 0 * H = 0 H = 0 H = 0 H = (-3) Snce s nomal to plane of polazaton of both and E, they have the H0 0 same plane of polazaton. Usng equaton (-3) we can smplfy the geneal equaton fo complex Poyntng vecto (equaton (-)) to the followng fom: S ( ) = exp ( ) (-4) H0 ωε and the aveage tme-doman Poyntng vecto s obtaned as: 6

27 S ωε (, t) = Re S( ) = exp( ) H ( ε ε ) 0 S (,t) s a functon of poston and t s paallel to f (-5) ε = 0. Smla to the pevous case we can detemne that medum s lossy, lossless o actve based on sgn of (,t) S. Usng equaton (-) the sgn of (,t) ( ). ε ε = ε. ε S s detemned by So the ctea fo type of popagaton (amplfyng, attenuatng, and lossless) n geneal depend on both medum and wave popagaton vecto: ω ε με ε ( ) ω ε με ε ( ) ω ε με ε ( ) Im > 0 amplfyng popagaton Im < 0 attenuatng popagaton Im = 0 lossless popagaton 3- Both and H ae NLP. Ths s the most geneal case and thee s no E0 0 smplfcaton possble. Also thee s no geometcal ntepetaton fo two nomal vectos that both ae nonlnea polazed. By compang equatons (-) and (-5) we conclude that fo waves wth NLP wave vecto, S (,t) depends on polazaton of and H but E0 0 accodng to equaton (-5), S (,t) s ndependent of and E0 H0 polazaton fo waves wth LP wave vecto. To nvestgate case 3, we need to use the TE/TM decomposton theoem. 7

28 In next secton we wll evew ths theoem and deve a fomulaton fo S (,t) and study ts mplcatons and applcatons..4 TE/TM Decomposton Accodng to TE/TM decomposton theoem f ( E,H) ae a soluton to souce fee Maxwell equaton n a smple medum, and û s an abtay eal unt vecto then we can expess ( E,H) as supeposton of two set of TE felds (, E H ) and TM felds ( E H ) that each one s a soluton to TE TE TM, TM Maxwell equatons n that medum and they have these popetes: E= E, ˆ TE + ETM ETE u = 0 H = H, ˆ TE + HTM HTM u = 0 Moeove all components of TE felds ( E, H ) can be expessed wth espect to a scala functon (feld) ψ ( ) and all components of TM felds TE ( E, H ) can be expessed wth espect to anothe scala functon ψ ( ). TM TM Both ψ TE ( ) and ψ TM ( ) satsfy Helmholtz equaton. In the specal case that thee s a decton TE TE TM û such that the ntal feld ( E,H) s ndependent of the pojecton of poston vecto n û decton (.e. ( ) ) (fo example f uˆ = yˆ then the feld ae ndependent of y and ae only functon of x and z) uˆ uˆ Then the scala functons ψ TE ( ) and ψ TM ( ) wll be the components of ( E,H) TM n û decton: ( ) E uˆ and ψ ( ) = H uˆ ψ = TM TE E and H wll be LP vectos paallel to û and gven as: TE 8

29 E = ( E uˆ) uˆ and H = ( H ) TM And the total feld can be wtten as: ( ) ( ) E= E uˆ uˆ+ E H = H uˆ uˆ+ H TE TM TE uˆ uˆ Now gven the plane wave wth felds ( ) ( ( ) = exp( j ) H H 0 we have: E = Eexp j ) and NLP wave vecto, f we choose û nomal to ( ) ( ) ( ) = uˆ uˆ + uˆ uˆ = uˆ uˆ whch shows s ndependent of TE/TM modes as follows: ( ) = exp( ) H ( ) = H exp( ) E E TM 0TM j E H ( E ) = uˆ uˆ 0TM 0 0TM E uˆ 0 = ωμ ( uˆ ) 0 and û and we can decompose the feld to TE 0TE j ( H ) H ˆ ˆ 0TE = 0 u u H uˆ 0 E0TE = ωε ( uˆ ) Now we fnd the elatonshp between total complex Poyntng vecto (S ) and complex Poyntng vecto of each mode ( S and S ) ( TE TM ) ( TE TM ) * * = = + + S E H E E H H * Ths can be smplfed to S= S + S + S TE TM coss * * E0u H0u * Scoss = ETE HTM = exp * ωμε E = E uˆ, H = H uˆ 0u 0 0u 0 ( ) ˆ u ( ) uˆ TE TM Theefoe S s a LP vecto paallel to û. coss 9

30 * If s LP then =0and theefoe S = coss 0 and supeposton s vald fo any decton û (whch s nomal to ). Ths can also be vefed usng equaton (-4). * * If s NLP then = j s a LP vecto paallel to and û s û ( ) neve zeo. So supeposton s only vald fo the dectons defned by eal unt vecto ˆv that vu ˆ ˆ = 0 because fo these dectons we have S vˆ = 0 and S vˆ= S vˆ+ S vˆ TE TM S t v= S t v+ S t vˆ (, ) ˆ (, ) ˆ (, ) TE TM And snce magnetc feld of TE mode s LP, fom equatons (-4) and (- 5) we obtan: STE = exp ( ) H0u ωε S TE 0u ωε (, t) = exp( ) H ( ε ε ) And snce electc feld of TM mode s LP, usng equaton (-) and (-) we have: STM = * exp ( ) E0u ωμ * S TM 0u ωμ (, t) = exp( ) E ( μ μ ) TE mode has a LP magnetc feld H and the TM mode has a LP electc feld E. S S S (, t) = (, t) + (, t) + Re( S ) TE TM coss S = S + S + S (, t) (, t) (, t) Re( ) uˆ = 0 uˆ = 0 S = 0 TE TM coss coss coss 0

31 Theefoe we have: S (, t) = S (, t) + S (,t ) TE TM If S (, t) < 0 and (, t) TE S < 0 TM then ( ) S,t < 0. In othe wods f a wave s attenuatng n both TE and TM modes then t s also attenuatng when both and H ae NLP. E0 0 In next secton we wll nvestgate the poblem of plane bounday between two smple medum. We wll defne an ncdent plane that contans the nomal to the bounday. If s LP then the ncdence plane s defned by nomal to the bounday and. If s NLP we assume that nomal to the bounday s paallel to polazaton plane of and call t ncdence plane. We choose the nomal to the bounday as z axs and the ncdence plane as xz plane (see fgues. and.5). So the poblem s D along y axs and the unt vecto uˆ = yˆ theoem as stated above. would satsfy the condton fo D TE/TM decomposton One of the mpotant esults that s useful fo multlaye meda s: zˆ S (, t) = zˆ S (, t) + zˆ S (,t) TE TM All possble cases of polazaton of, and H ae shown n table -. E0 0 Table.: All possble polazatons fo paametes of a plane wave E 0 H TE 0 y /TM y decomposton LP LP LP TE y o TM y o TE y +TM y LP NLP NLP TE y +TM y NLP LP NLP TM y NLP NLP LP TE y NLP NLP NLP TE y +TM y

32 .5 Geometcal Repesentaton of Complex Wave Vecto fo Dffeent Types of Meda Usng equaton (-7) we can daw dffeent confguatons of complex wave vecto n a gven coodnate system. We wll use these epesentatons late n study of multlaye meda. Snce a NLP (nonlnea polazed) complex vecto defnes a plane we can choose ths plane to be xz plane wthout loss of genealty (WLOG) Fo a medum of type I dffeent possbltes ae shown n Fgue.. The blac vectos ae eal pat and the ed vectos ae magnay pat of complex vectos. z A B C D x G E F H Fgue.: Dffeent confguatons of complex wave vecto n type I medum Snce late we wll consde multlaye meda that ntefaces ae nomal to z- axs, we don t need to consde the cases that ae symmetc to z-axs n Fgue.

33 Cases A and E ae LP and othe cases ae NLP. The possble cases fo type II meda ae vey smla to type I except that > and =0 fo lnea polazaton. These ae shown n Fgue (.): z A B C D x G E F Fgue.: Dffeent confguatons of complex wave vecto n type II medum H Fo type III medum we have = and < 0. Dffeent cases ae depcted n fgue (.3) In ths fgue cases A and E ae LP and othe cases ae NLP. 3

34 z A B C D x E F G H Fgue.3: Dffeent confguatons of complex wave vecto n type III medum Fo type IV medum we have = and > 0. Dffeent cases ae llustated n fgue (.4). In cases A and E the complex wave vecto s LP and n othe cases t s NLP. Ths type meda s actve because fo example when the complex vecto s LP accodng to equaton (-5) the aveage tme-doman Poyntng vecto flows n the same decton ( ) that ts ampltude nceases ( ) whch s chaactestc of an actve medum. 4

35 z A B C D x E F G H Fgue.4: Dffeent confguatons of complex wave vecto n type IV medum 5

36 Chapte 3 Complex Plane Wave Soluton of Maxwell Equatons n Pesence of Plane Boundaes 3. Complex Wave Vecto Matchng In ths chapte fst we nvestgate the plane wave soluton of Maxwell equatons n a plane bounday between two smple medum. A plane wave soluton n medum s consdeed as ncdent wave o exctaton. The z axs s the nomal to the plane bounday between two smple medum wth gven pemttvty and pemeablty. The ncdent wave (whch s soluton of Maxwell equatons fo a smple medum wthε, μ ) s epesented as E( ) = E0exp( j ). The complex wave vecto can be LP o NLP. If t s LP we use ts polazaton to defne the ncdence plane (xz). If t s NLP we assume that z axs s paallel to ts polazaton plane and so the xz wll be the polazaton plane of t wll not have any Y-component. We defne two (unnown) plane waves that togethe wth the ncdent wave ae gong to satsfy both Maxwell equatons and bounday condtons., so 6

37 z ε, μ Medum x ε, μ Medum Fgue 3.: Two smple medum wth plane bounday between them The unnown plane wave n medum and s called eflected wave and s also a soluton of Maxwell equatons n ths medum: ( ) = exp( j ) E E n whch and ae unnown. E03 3 Anothe unnown plane wave (tansmtted o efacted wave) s defned n medum and s epesented as E ( ) = E exp( j ) n whch and 0 E0 ae unnown. Ths s a soluton of Maxwell equatons n medum. Contnuty of tangental electc feld at the bounday gves us ths equaton: ( ) ( ) ( ) zˆ E exp j + zˆ E exp j = zˆ E exp z = (3-) The necessay condton fo equaton (3-) to hold tue fo = 0s: = = z = 0 (3-) Equaton (3-) s vald egadless of the polazaton of vectos E 0m, m =,, 3. Fo (3-) to be tue t s necessay that tangental components of the thee complex wave vectos: 7

38 xˆ = xˆ = xˆ 3 (3-3) Thee s no estcton to the polazaton of wave vectos. Assumng = xˆ + zˆ, m=,, 3 m mx mz It s easy to fnd the wave vecto of eflected wave: = = ωμε + = + z z 3 3 x z 3x 3 (3-4) Fom equatons (3-3) and (3-4), we conclude that: 3z = (3-5) Geometcal ntepetaton of equaton (3-5) s that s symmetc of 3 wth espect to x-axs. Theefoe 3 wll have the same polazaton as (LP o NLP). Fo fndng we have: = ωμε = =± ωμε x (3-6) x x z The coect banch of z can be selected usng decton of flow of powe. Snce ncdent plane s n medum decton of flow of powe s fom meda to meda and ths means z ˆ S (, t) s negatve. It s vey mpotant to notce that thee s no estcton to polazaton of. can be LP o NLP egadless of polazaton of as long as the contnuty of tangental component of complex wave vecto acoss the bounday ( = ) s satsfed. x x 8

39 3.. Ctea fo Desgnatng Waves as Incdent, Reflected and Tansmtted (Refacted) When we ae dealng wth lossless delectcs t s staghtfowad to desgnate ncdent, eflected and tansmtted waves because the decton of flow of enegy s paallel wth the wave vecto. As t was shown n equaton (-7) the vectos S (,t) and ae paallel. That s not the case when we ae dealng wth meda that have negatve o complex pemeablty and pemttvty. In fgue (3.) we assume that flow of powe s fom meda to meda. So we wll have ncdent and eflected waves n medum and tansmtted wave n medum. Nomal components of complex Poyntng vecto and tme aveage Poyntng vecto ae contnuous acoss the bounday. Theefoe t s consstent that we use z ˆ S (, t ) as ctea. Fo ncdent wave must be towads z ˆ S (, t) bounday, so n fgue (3.) S ( t ) s negatve fo ncdent wave. Smlaly ˆ (, ) z ˆ, z S t > 0 eflected wave we use the condton zˆ S (, t) > 0 and fo tansmtted wave ( t) zˆ S, < 0. Fo example f medum s a double negatve metamateal and medum s a lossless delectc, ncdent, eflected and tansmtted waves ae shown n fgue (3.). fo 9

40 Usng equaton (-5) we have zˆ. S (, t) Aε zˆ. n whch A >0. Usng ths = elatonshp we obtan the wave desgnaton based on the above-mentoned pocedue. ε = ε < 0 μ = μ < 0 3 Reflected 3 Incdent ε = ε > 0 μ = μ > 0 Tansmtted Fgue 3.: Detemnng ncdent eflected and tansmtted waves usng powe flow cteon In fgue (3.) s ncdent wave, s eflected wave and s the tansmtted wave Geometcal epesentaton of Complex Wave Vectos n Plane Bounday Poblem In ths secton we use the esults of pevous secton to dffeent meda types ntoduced n secton (.). Snce thee ae fou types of meda, t esults n 6 dffeent plane bounday poblems. 30

41 We wll consde seveal cases that ae of moe pactcal mpotance, n the followng subsectons. 3.. Plane Bounday between Two Medum of Type I Ths s the most famla case of a plane bounday between two lossless delectcs. It s mpotant to notce that nowng the type of medum doesn t gve us enough nfomaton to detemne the appopate sgn n equaton (3-6). Fo that pupose we need to now some nfomaton about pemttvty and/o pemeablty. Fgue (3.3) shows all possble cases of dffeent wave vectos that can be matched when we have a plane bounday between two medum of type I. Real vectos ae shown n blac and magnay vectos ae shown n ed. 3

42 z Medum Type I case case case 4 case 3 case 5 x Medum Type I Fgue 3.3: Two smple medum of type I wth plane bounday between them In fgue (3.3) each of medum o medum can be chosen as the medum that contans the ncdent wave and any wave vecto n a symmetc pa can be selected as ncdent wave, then the othe wave vecto wll be the ncdent wave. Any pa of wave vectos n medum o can be selected as ncdent/eflected waves. The wave wth z ˆ S (, t) towad bounday s ncdent wave and the wave wth z ˆ S (, t) away fom bounday s eflected wave. If z ˆ S (, t ) =0 then we use (,t) S (,t) eflected f S (,t) S as ctea. If deceases towads bounday the wave s ncdent and the wave s deceases away fom bounday. Afte detemnng ncdent and eflected wave n one of the medums, we must choose the coect banch among two possbltes. If zˆ S (, t) s 3

43 away fom bounday we choose that wave as tansmtted wave. If z ˆ S (, t ) =0, we use S (,t) as ctea and the wave wth deceasng S (,t) away fom bounday wll be the tansmtted wave. We apply the above mentoned ctea to seveal cases. Assumng that medum s a meda of type I wth ε = ε such that ε > 0 and meda s also of type I wth ε = ε such that ε < 0. We want to detemne the desgnaton of waves n case shown n fgue (-6). Usng equaton (-7) we have : ε S (, ) H (3-7) t = 0 ωε whch ndex s used fo ncdent wave wth ctea zˆ S (, t) < 0 whch s equvalent to ε zˆ > 0 and snce ε > 0 t s equvalent to zˆ < 0. Smlaly the eflected wave wth ndex 3 s ε S (, ) H (3-8) 3 t = 03 3 ωε and the condton zˆ (, t) S 3 > 0 s equvalent to zˆ 3 > 0. Fo the tansmtted wave wth ndex n meda we have ε S (, ) H (3-9) t = 0 ωε and zˆ (, t) S < 0 s equvalent to ε ˆ z < 0 whch means zˆ > 0 The esult s shown n fgue (3.4) both n wave vecto epesentaton and ay epesentaton. 33

44 ε = ε > 0 μ = μ > 0 3 Incdent Reflected ε = ε < 0 μ = μ < 0 Tansmtted Fgue 3.4: ncdent, eflected and tansmtted wave at nteface between a lossless delectc and lossless DNG metamateal (both meda ae type I) Anothe obsevaton n fgue 3.4 s that the same ncdent wave n one medum can excte dffeent tansmtted waves n the othe medum. Fo example n cases and the ncdent wave s a wave wth LP (eal) wave vecto n medum that can excte a LP wave vecto n (case ) o NLP wave vecto (case ) n medum. Case s the famla phenomenon of total eflecton and the LP (eal) wave vecto n medum exctes the NLP wave vecto n medum. Fo a lossless delectc equaton (-) o (-5) shows that decton of flow of powe s ˆx and t exponentally decays n ẑ decton, whch s the well nown evanescent o suface wave. Due to ecpocty f n case we consde the evanescent waves n medum as ncdent wave, the tansmtted wave wll be a LP wave wth eal wave vecto. 34

45 3.. Plane Bounday between Medum of Type I and II All dffeent possbltes ae llustated n fgue 3.5. The blac vectos ae eal pat of complex wave vectos and the ed vectos ae magnay pat of complex wave vectos. z case 3 case 4 case case case 5 case 6 case 7 Medum Type I x Medum Type II Fgue 3.5: Two smple medum of type I and II wth plane bounday between them We obseve that n cases and the same ncdent wave n medum could have dffeent tansmtted waves dependng on the actual value of paametes. The same thng happens n cases 3 and 4 when the ncdent waves s n medum ae the same but the they can excte dffeent tansmtted waves n medum. 35

46 We choose case and apply the pocedue explaned n pevous secton to desgnate ncdent, eflected and tansmtted waves (Fgue 3.6) z ε = ε > 0 μ = μ > 0 3 Reflected Incdent x ε = ε < 0 μ = μ > 0 Tansmtted Fgue 3.6: Plane bounday between a lossless delectc and a lossless SNG metamateal (Plasmons) Thee s no flow of powe n z decton to medum and the tansmtted wave s the so-called suface Plasmon. The Poyntng vecto of the tansmtted wave s paallel to x axs but ts decton depends on polazaton of the ncdent wave. Fo TM y mode usng equaton (-) we have S (, t) = exp ( ) μ E0 (3-0) ωμ and fo TE y usng equaton (-5) we have S (, t) = exp ( ) ε H0 (3-) ωε By compang (3-0) and (3-) we obseve that n TM y mode powe flows n ˆx decton but fo TE y mode powe flows n ˆx decton. Ths s the 36

47 dffeence between suface Plasmon and the suface wave (evanescent wave) that s fomed between two delectcs, although they have the same wave vectos Plane Bounday between Medum of Type I and III All dffeent possbltes ae llustated n fgue 3.7. The blac vectos ae eal pat of complex wave vectos and the ed vectos ae magnay pat of complex wave vectos. z case case case 3 case 4 case 5 Medum Type I NLP L L x Medum Type III LP Fgue 3.7: Two smple medum of type I and III wth plane bounday between them 37

48 We consde case when meda s a lossless delectc wth ε = ε and μ = μ (fo example fee space) and meda s a delectc wth hgh loss wth ε = jε and μ = μ. (Fgue 3.8) The ncdent wave s n medum so we need to detemne the coect banch of tansmtted wave n medum. Fo that we need t calculate S (,t) whch depends on TE/TM mode (polazaton). Fo TM y mode usng equaton (-) we obtan: S E ωμ (, t) = exp( ) 0 Fo TM y mode we must choose μ (3-) such that zˆ < 0. Fo TE y mode usng equaton (-5) we have: S (, t) = exp( ) H ( ε ) (3-) 0 ωε Fo TE y mode we have to select so that zˆ > 0. By nvestgatng case n fgue (3.5) we see that fo both modes the same banch can be selected fo tansmtted wave. Fo a hghly lossy delectc ε >> we have >> because accodng to = = ω με So the vecto ( ) (3-3) wll be vey close to nomal to the bounday and snce n the TE y mode the popagaton s n decton the flow of powe n meda s neglgble and powe flow densty emans constant along the bounday. 38

49 z ε = ε > 0 μ = μ > Reflected Incdent x ( ) ε = jε, ε > 0 μ = μ > 0 Tansmtted Fgue 3.8: Plane bounday between a lossless delectc and a hghly lossy delectc (Zennec Waves) Fo TM y polazaton we have R TM y E = = E y3 y z z μ μ z z + μ μ (3-4) Assumng both meda ae non-magnetc mateals we have μ = μand R =, >> R (3-5) z z TM y z z TM z+ z So fo TM y polazaton (whch s usually efeed as TE z n lteatue) total feld close to bounday s small. 39

50 Fo TE y polazaton we have R TEy z z H y3 ε ε = = H y z z + ε ε (3-6) z In ths case we can show that << z then RTE and the total feld s ε ε y maxmum close to suface. Ths popety s used fo low fequency boadcastng and sgnal eaches to ponts beyond the lne of sght, because at those fequences ant antenna ove eath can be consdeed as to be a pont close to bounday. And the fact that s close to nomal to the bounday shows that S (,t) deceases slowly along the bounday. 3.3 Poyntng vecto of supeposton of ncdent and eflected waves To nvestgate the flow of powe n mult-laye meda t s essental to calculate the Poyntng vecto of the total feld esulted as supeposton of ncdent and eflected feld. Fo ths pupose we consde the TM and TE modes sepaately. y y 40

51 3.3. Supeposton of ncdent and eflected wave n mode TE y To be consstent wth the notatons that we used n eflecton and tansmsson fom a plane bounday, we use ndex fo ncdent wave and ndex 3 fo eflected wave. Ou goal s to calculate the total Poyntng vecto wth espect to paametes of ncdent and eflected plane waves. Assumng that xz s the ncdence plane, we have: ˆ ˆ = x x + z z and ˆ ˆ 3 = x x z z H = H yˆ and = H yˆ, ( ) ( ) H j H0 H0 e ϕ j 3 =, H = H e ϕ * * * * S= E +E 3 H +H3 = S+ S3 + E H3 + E 3 H (3-7) S S = * ( ) * 3 H0H03 exp x x jz z H0H03 exp( xx jz z 3 ωε + + ) In whch and S ae complex Poyntng vecto of ncdent and eflected S 3 wave espectvely and wee nvestgated n secton (-3). The tem S 3 s esult of the coss-tem that esults fom coss poduct of ncdent electc feld ( ) wth eflected magnetc feld ( H ) and coss poduct of ncdent E 3 H E3 3 magnetc feld ( ) wth eflected electc feld ( ). We call S cosstem. So we have establshed a coespondence between decomposton of total feld to ncdent and eflected components and decomposton of total Poyntng vecto. 4

52 Usng equaton (-4) we have yˆ S = 0, yˆ S 3 = 0 and fom (3-7) we have yˆ S = 0. So the total Poyntng vecto also doesn t have any component 3 nomal to plane of ncdence: ŷ S = 0 We ae nteested n nomal component of tme-aveage Poyntng vecto whch can be obtaned usng equaton (-5): H zˆ. S, t exp x z ωε 0 ( ) = ( x + z ) ( ε z ε z) H03 zˆ. S (, t) = exp ( x z) ( ε + ε ωε 3 x z z z ), = Re( ), = Im( ) z z (3-8) z z And the mutual nteacton tem S afte some smplfcaton s obtaned as: 3 H0H03 zˆ Re( S ) = exp( x) sn ( z+ ϕ ϕ )( ε ε z ) (3-9) 3 x z 3 z ωε So we can wte the nomal component of total feld as sum of the nomal components of ncdent and eflected wave and a thd tem whch s esult of nteacton of electc feld of ncdent wave wth magnetc feld of eflected wave and magnetc feld of ncdent wave wth electc feld of eflected wave. zˆ. S(, t) = zˆ. S(, t) + zˆ. S3(, t) + zˆ Re( S 3) (3-0) Thee ae seveal specal cases that ae of pactcal mpotance. If we have ncdent and eflected waves wth NLP wave vecto n a lossless delectc as shown n case 3 n fgue -7, then we have * ˆ ˆ = xx+ jz z and = ˆ ˆ x x jzz = and ε, μ 3 (3-) 4

53 H0 S (, t) = exp( zz) ωε x H S3 t zz ωε 03 x ˆ and (, ) = exp( ) x xˆ H H Re cos x sn { ˆ x z z} 0 03 ( S ) = ( ϕ ϕ ) ˆ + ( ϕ ϕ ) ωε (3-) The total Poyntng vecto s obtaned as: x S(, t) = H0 exp( ) 03 exp( ) 0 03 cos( 3 ) ˆ zz + H zz + H H ϕ ϕ x ωε H0H03 z sn ( ϕ3 ϕ) z ˆ ωε + (3-3) So the total tme-aveage Poyntng vecto contans an x component whch s a functon of z and a z component whch s constant: zˆ. S, t = ( ) z S t = 0 0 and ˆ. (, ) H H zˆ S, t = z sn ϕ3 ϕ ωε 0 03 ( ) ( ) 3 (3-4) So we have shown although ncdent wave and eflected wave ndvdually don t tansfe any powe n z decton, the supeposton tansfes powe n the z decton. We can use these esults to show that how powe s coupled between two delectcs wth an a gap between them as shown n fgue 3.9. It s assumed that the ncdence angle of ncdent wave n medum s geate than ctcal angle so that the wave n the a gap s evanescent. 43

54 S (, t) s a constant vecto n medums and 3. In medum zˆ S(, t) constant but xˆ S(, t) Snce zˆ S(, t) s a functon of z, as shown by equaton (3-3). s contnuous acoss the bounday we conclude that t s s constant and equal n all thee meda and powe tansfes n z decton fom laye to laye 3. (Fgue 3.9) Although the polazaton of wave vecto and polazaton of magnetc feld change fom lnea to nonlnea when the wave entes nto a gap, the tansmtted wave to medum 3 s a plane wave wth LP wave vecto and LP magnetc feld. Ths popety s used n optcs to sample the optcal sgnal fom a delectc wavegude by a psm. z Wave vectos Total Poyntng vectos S, t ( ) ε > 0 μ > 0 ε > 0 μ > 0 3 ε > 0 3 μ > 0 3 x Fgue 3.9: The a gap between two delectcs wth wave vectos and aveage tme Poyntng vectos. The z component of Poyntng vectos n all thee meda s constant and equal. We have exactly the same stuaton as shown n Fgue 3.9 when medum s a slab of sngle negatve metamateal n a ( o suounded by othe 44

55 delectcs.e. medum and 3 ae a o any othe delectc). The only dffeence s that we have NLP wave vecto n the medum egadless of the ncdent angle of the plane wave n medum. The case fo nomal ncdent wave to a slab of sngle negatve metamateal has been shown n fgue 3.0. The total Poyntng vecto n the slab s: H H S t z ϕ3 ϕ ωε (, ) = 0 03 sn( ) z ˆ (3-5) whch s constant n the slab and equal to Poyntng vecto n medums and 3. z Wave vectos Total Poyntng vectos S, t ( ) ε > 0 μ > 0 ε < 0 μ > 0 3 ε > 0 3 μ > 0 3 x Fgue 3.0: Nomal ncdence on a sngle negatve metamateal slab. All Poyntng vectos ae constant and equal. 45

56 Anothe specal case s supeposton of ncdent and eflected felds wth LP wave vecto n a lossless delectc. In ths case we have 3 = xˆ + zˆ x z = xˆ zˆ x z and ε, μ H0H03 Re( S ) = cos( z z+ ϕ ϕ ) x xˆ 3 3 ωε Usng (3-7) we have: (3-6) S t H H H H z x H H ωε ωε x ( ) z, = ( ) ˆ ( cos z + ϕ3 ϕ ) Ths shows that the mutual nteacton tem zˆ Re( S ) s equal to zeo and also zˆ. S(, t) s a constant but xˆ. S(, t) 3 s a functon of z. zˆ zˆ. S, t zˆ. S, t zˆ. S, t H H ωε ( ) = ( ) + 3( ) = z ( 0 03 ) In the case of total eflecton whch absolute value of eflecton coeffcent s equal to one, we have H0 = H03 and zˆ. S(, t) = 0 whch econfms the contnuty of nomal component of Poyntng vecto, consdeng the fact that the z-component of Poyntng vecto of suface wave s also zeo. Ths explans why a slab of delectc n a gudes the powe n x-decton as shown n fgue

57 z ε > 0 μ > 0 Wave vectos Total Poyntng vectos S, t ( ) ε > 0 μ > 0 3 ε > 0 3 μ > 0 3 x Fgue 3.: A delectc slab n the a, wth wave vectos and total tme-aveage Poyntng vectos. In ths case thee s no flow of powe nomal to the boundaes. It s also woth mentonng that, the z component of total Poyntng vecto (nomal to the boundaes) s constant n both (3-3) and (3-6). Ths guaantees the lossless flow of powe fom fst sem-nfnte medum to all mddle layes to the most bottom sem-nfnte medum whch s also consstent wth contnuty of nomal component of Poyntng vecto acoss the bounday Supeposton of ncdent and eflected wave n TM y mode Smla to TE y mode we can calculate the total Poyntng vecto of the supeposton of the ncdent and eflected wave. ˆ ˆ = x x + z z and ˆ ˆ 3 = x x z z E = E yˆ and = E yˆ, E j E0 E0 e ϕ (3-7) j 3 =, E = E e ϕ

58 ( ) ( ) S= E +E 3 * * * * H +H3 = S+ S3 + E H3 + E 3 H (3-8) S 3 In whch and S ae complex Poyntng vecto of ncdent and eflected S 3 wave espectvely. The tem S s esult of nteacton of ncdent electc feld ( ) and eflected magnetc feld ( H ) and nteacton of ncdent E 3 magnetc feld ( H nteacton tem. 3 ) wth eflected electc feld ( E ). We call S mutual 3 3 Nomal components of and S can be calculated usng equaton (-) S 3 E zˆ. S, t exp x z ωμ 0 ( ) = ( + ) ( μ μ ) x z z z, = Re( ), = Im( ) z z z z E03 zˆ. S3(, t) = exp ( xx zz) ( μ z + μ z) (3-9) ωμ And the mutual nteacton tem S 3 afte some smplfcaton s obtaned as: E0E03 zˆ Re( S 3 ) = exp ( xx) sn ( zz+ ϕ3 ϕ )( μ z + μ z ) (3-30) ωμ And the nomal component of the total Poyntng vecto can be expessed as zˆ. S, t zˆ. S, t zˆ. S3, t zˆ Re ( ) = ( ) + ( ) + ( ) S 3 (3-3) The same aguments that we made fo specal stuctues n TE y mode ae vald also fo TM y mode. 48

59 3.3.3 Supeposton of ncdent and eflected wave n mxed mode We can decompose the ncdent, eflected and total feld to TE/TM components. Assumng that xz s ncdent plane we can decompose the total feld to TE/TM components as dscussed n secton.4: E= E ˆ y y+ E H = H ˆ y y+ H TE TM (3-3) The total Poyntng vecto s obtaned as: S= S + S + E H * TM TE TE TM * S = E yˆ H and S = E H * yˆ TM y TM TE TE y * snce the tem E H s paallel to ŷ we can wte: TE TM (3-33) zˆ S= zˆ S + zˆ S TM TE zˆ S t = zˆ S t + zˆ S t (, ) (, ) (, ) TM TE (3-34) usng the esults that we obtaned n 3.. and 3.. we can expess S TM and S TE wth espect to eflected and ncdent waves: S = S + S + S S = S + S + S TM TM 3TM 3TM TE TE 3TE 3TE (3-35) zˆ S t = zˆ S t + zˆ S t + zˆ zˆ S t = zˆ S t + zˆ S t + zˆ (, ) (, ) (, ) Re( S ) TM TM 3TM 3TM (, ) (, ) (, ) Re( S ) TE TE 3TE 3TE (3-36) Addng the equatons togethe we have: zˆ S(, t) = zˆ S (, t) + zˆ S3 (, t) + zˆ Re( S3 TM + S 3TE ) (3-37) 49

60 Assumng H = H ˆ y and H = Hy 3 3ˆ ae magnetc felds of ncdent and eflected waves n futhe smplfed to: y TE y mode and E = Eyˆ y and E3 = Ey3ˆ y ae electc felds of ncdent and eflected waves n TM mode equaton (3-33) can be ( )( 3 3) Re( )( 3 3 ) E H = yˆ jim H E H E + H E H E * * * * * * * TE TM x z y y y y x z y y y y * The tem E H has ŷ component, and ths s contast to total complex TE TM Poyntng vecto of pue TE/TM modes ( S TM and S TE ) that don t have any ŷ component. Ths s because both E TE and HTM so the coss poduct wll be n y ˆ decton. y (3-38) ae pependcula to ŷ and We use the esults that we have obtaned to ths pont to compae wave vectos, tme-aveage Poyntng vecto and magnetc felds (n TM y mode) fo plane wave ncdent on nteface of two lossless delectcs fo two cases of patal and total eflecton. It s vey mpotant to notce that complex Poyntng vecto of pue modes S TM and S TE ae n xz plane, n othe wods fom (3-7) and ŷ S = 0 TE and (3-8) we have: yˆ S = 0, whle the mxed mode Poyntng vecto gven n TM * e TE TM quaton ( 3-33) the coss tem E H s n ŷ decton. 50

61 z ε = ε > 0 μ = μ > 0 Wave vectos 3 Poyntng vectos S, t S ( ) ( t) 3, S S ( t), (, t) Magnetc felds of TM y mode H 03 H 0 x ε = ε > 0 μ = μ > 0 S ( t), H 0 Fgue 3.: wave vectos, tme aveage Poyntng vecto and magnetc feld n patal eflecton fom nteface of two lossless delectcs Wave vectos Poyntng vectos S, t ( ) Magnetc felds of TM y mode z S ( t) 3, H 03 ε = ε > 0 μ = μ > 0 3 S ( t), S (, t) H 0 H 03 x ε = ε > 0 μ = μ > 0 S ( t), H 0 H 0 Fgue 3.3: wave vectos, tme aveage Poyntng vecto and magnetc feld fo total eflecton, fom nteface of two lossless delectcs 5

62 Assumng ( ) E = E0exp j yˆ and E0 s eal (wthout loss of genealty) and consdeng the eflecton coeffcent: R TM y z z E03 μ μ = = E 0 z z + μ μ (3-39) We notce that R TM y s eal when patal eflecton happens and s complex wth modulus equal to. Thee ae seveal emas that we can mae by compang total eflecton and patal eflecton as shown n fgues 3. and H03 s LP n both cases but t s eal (co-phase wth ncdent wave) n patal eflecton and t s complex n total eflecton. The total magnetc feld n medum fo total eflecton s NLP. So the dffeence between total eflecton and patal eflecton s not just n efacted wave. - 0 H s LP n patal eflecton but t s NLP n total eflecton 3- Thee s flow of powe n z decton n patal eflecton but thee s no flow of powe n z decton when we have total eflecton. 4- S ( t) s exponentally decayng when thee s total eflecton n, contast to the patal eflecton whch s a constant vecto. 5

63 3.4 Plane waves n plasma and suface plasmon and the popetes The fomulaton that we have developed to ths pont wll be used to study the popetes of nteface between a delectc and plasma. The analyss hee s lmted to classcal Maxwell s equaton and modelng of mateals n each laye as smple medum (lnea, homogeneous, and ansotopc). We wll not consde quantum effects and othe physcal phenomena beyond ths model. Fo example the fact that Plasmon s only ceated n the suface s mode led by a vey thn homogeneous laye wth negatve pemttvty. Plasma s defned as a delectc wth pemttvty ε = ε jε wth 0 ε <. The physcal mplementaton and fequency dependence of ths pemttvty s beyond the scopes of ths model Suface plasmonc esonance Suface plasmon as fa as ou model s concened, s a thn laye of plasma n the metal close to ts nteface wth a delectc. Although we wll analyze nteface of a bul delectc and bul plasma, but the fact that feld decays exponentally as we move fa fom nteface, justfes ths assumpton. Fgue 3.4 shows nteface of a lossless delectc wth lossless plasma. Ths s case shown n fgue 3.5. Total tme-aveage Poyntng vecto n meda can be obtaned fom equaton (3-3) n whch snce eflecton coeffcent 53