Propagation of Current Waves along Quasi-Periodical Thin-Wire Structures: Accounting of Radiation Losses

Transcription

1 Intaction Nots Not 6 3 May 6 Popagation of Cunt Wavs aong Quasi-Piodica Thin-Wi Stuctus: Accounting of Radiation Losss Jügn Nitsch and Sgy Tkachnko Otto-von-Guick-Univsity Magdbug Institut fo Fundamnta Ectica Engining and Ectomagntic Compatibiity Abstact Th homognous pobm of cunt popagation aong a thin wi of abitay gomtic fom na gound is ducd us of th Fu-Wav Tansmission Lin Thoy [8-] to a Schöding-ik diffntia quation, with a potntia dpnding on both th gomty of th wi and fquncy. Th potntia is a compx vaud quantity that cosponds to ith adiation osss in th famwok of ctodynamics o to th absoption of patics in th famwok of quantum mchanics. If th wi stuctu is quasi piodica, i.., it consists of a finit numb of idntica sctions, th potntia can b appoximaty psntd as a st of piodicay aangd idntica potntias. W us th fomaism of tansf matics and find an anaytica xpssion fo th tansmission cofficint of th finit numb of piodicay ocatd non-unifomitis which aso contains th scatting data of on nonunifomity. Th obtaind sut yids th possibiity to invstigat fobiddn and aowd fquncy zons which a a typica fatu of piodic stuctu. This wok was sponsod by th Rsach Institut fo Potctiv Tchnoogis and NBC Potction WIS Munst und th contact numb E/E59/3X45/F37.

2 Contnt. Intoduction.3. Equation fo th cunt in a wiing systm with osss.4.. Mixd-Potntia Intga Equations MPIE fo th wiing systm with osss Th scond MPIE quation: Bounday condition fo th coatd wi Th fist MPIE quation: Bounday condition fo th finit conductiv wi 7. Fu-Wav Tansmission Lin FWTL quations fo th wiing systm with osss. Itation appoach fo th goba paamts Popagation of Cunt Wavs aong Quasi-Piodica Wiing Stuctus: a Quantum Mchanica Anaogy FWTL and Schöding-ik quation On-dimnsiona scatting pobm. Rfction and tansmission cofficints fo ossy systms Tansf matics fo th on-dimnsiona scatting pobm Tansf matix fomaism fo th quasi-piodica systm with osss. 3.5 Aowd and fobiddn fquncy zons. Connction of th paamts fo quasipiodica and piodica systms 9 4. Numica xamp. Rativ impotanc of th diffnt ossy mchanisms.3 5. Concusion 46 Rfncs.47 Appndix: Numica soution of th Schöding-ik quation by th mthod of tansf matics.48

3 I. Intoduction An anaysis of popagation of cunt wavs aong piodica stuctus in diffnt adiotchnica and cto-tchnica appications bcoms ncssay. Ths piodica stuctus show non-tivia ctodynamica poptis and somtims can b usd to buid fits, antnnas and HPM soucs. Moov, bcaus of th simpicity of buiding piodica thinwi stuctus, thy can sv as mnts fo th dsign of mta-matias. Th piodica infinit stuctus a studid in a numb of paps [,] fom an ctodynamica point of viw. Th popagation of cunt aong infinit piodica tansmission ins aso can b appid to th piodica xcitation of mchanica systms paamtica sonancs [3]. Howv, in a if on das with quasi-piodica systms: systms, which consist of a finit numb of idntica sctions. In th pvious paps [4,5] w invstigatd th popagation of cunt wavs aong a wi without any osss adiation, ohmic, dictic, tc.. W tansfomd th quations of nonunifom tansmission in thoy with ngth dpndnt inductanc and capacitanc p-unit ngth to a scond od diffntia quation fo som auxiiay function, which is simpy connctd with th cunt. Th quation ooks ik a usua on-dimnsiona Schöding quation in quantum mchanics, which potntia can b cacuatd using th p-unit ngth tansmission in paamts. Th potntia dcays to zo at pus and minus infinity and th ngy of th patic is positiv, thus w da with a on-dimnsiona quantum mchanica scatting pobm [5] and can appy powfu and w-dvopd mathmatica mthods to invstigat such pobms. Fo th quasi-piodica wiing systm th potntia is aso quasi-piodica. Using th fomaism of th tansf matix [6] w find an anaytica xpssion fo th tansmission cofficint of th finit numb of piodicay ocatd nonunifomitis which aso contains th scatting data fo on non-unifomity. pnding on th fquncy th absout vau of th tansmission cofficint osciats. This cosponds to fobiddn and aowd fquncy zons which a typica fo piodica stuctus [7]. Howv, th usuay usd wiing systms hav diffnt kinds of osss: dictic osss, ohmic osss, and adiation osss. In th psnt pap, w gnaizd th fomaism [4,5] fo such systms. W consid quasi-piodica stuctus, which consist of a wi with finit conductivity coatd by ossy dictic insuation. Fo this systm, fo high fquncis, whn adiation osss can bcom substantia, th simp appoach to a non-unifom tansmission in usd in [4] is not appicab, and w hav to appy a gna Fu Wav Tansmission Lin thoy FWTL [8,9,], which has to b modifid fo th cas of dictic and ohmic osss. Fom th fist stp in th scond Sction of this pot, w obtain bounday conditions fo th fist and scond Mixd Potntia Intga Equation MPIE, as fo th potntia, as w as fo th tangntia ctica fid on th bounday of th wi. Thn w us th obtaind MPIE and gna tchniqus dscibd in [] to div simp anaytica xpssions fo th goba paamts of th Fu-Wav Tansmission Lin thoy. Using th goba paamts th FWLT quations a ducd to th Schöding-ik quation fo som auxiiay function connctd with th cunt in a simp way. Howv, now th potntia in th Schöding-ik quation is, in contast to th ossss cas, compx-vaud. Th xpicit connction btwn th osss and th imaginay componnt of th potntia is stabishd at th bginning of th thid Sction. Haft, w gnaiz th fomaism of th tansf matix fo ossy systms and div an anaytica xpssion fo th tansmission cofficint of th finit numb of piodicay ocatd ossy non-unifomitis, which aso contains th scatting data fo on non-unifomity. Nxt, th 3

4 connction btwn th tansfomation paamts of quasi-piodica systms and paamts of cosponding piodica systms quasi-pus is stabishd. In th fouth Sction, w consid a numica xamp of th quasi-piodica in with ohmic, dictic and adiation osss. It wi b shown, that th zon stuctu, which xists in th ossss cas hods aso fo th ossy cas. Again, in dpndnc on th fquncy th absout vau of th tota popagation cofficint though th chain osciats. That cosponds to fobiddn and aowd fquncy zons. Fo th considd xamp, both dictic and ohmic osss a ngigib; howv, th adiation osss dcas ssntiay th popagation cofficint though th chain fo th aowd fquncy zons. In concusion, w fomuat som dictions fo futu invstigations.. Equation fo th cunt in a wiing systm with osss... Mixd-Potntia Intga Equations MPIE fo th wiing systm with osss. In this Sction, w obtain th Mixd-Potntia Intga Equations MPIE fo th coatd wi with ohmic osss, which at wi b a basis fo th vauation of th FWLT quations with goba paamts. Th systm of MPIE fo a pfcty conducting uncoatd thin wi consists of two intga quations [] which coup th tota tangntia cunt inducd in th wi, and th potntia w us th Lonz gaug on th bounday of th wi. Th fist quation is a zo bounday condition fo th tangntia componnt of th tota ctic fid xciting xtna fid pus th scattd fid gnatd by th inducd cunt. Th scond on is just an intga xpssion fo th scaa potntia in th Lonz gaug in tms of th tota tangntia cunt inducd in th wi. It is obvious, that both of thm hav to b changd fo th coatd wi with finit conductivity. Bcaus ths ffcts tun out to b sma, w wi tat thm spaaty: i.., fo th fist quation w consid an uncoatd wi with a finit conductivity, and fo th scond on w consid th coatd pfcty conducting wi... Th scond MPIE quation: Bounday condition fo th coatd wi. In this Sction, w woud ik to obtain an xpssion fo th potntia on th bounday of th mtaic kn of th coatd wi if th potntia on th bounday of th uncoatd wi is known. Lt us consid th scatting by a thin dicticay coatd wi. A unifom coating of thicknss b a is pacd ov a pfcty conducting wi with adius a. W assum, that th wi is thin and smooth, namy that b << λ,/ K, wh λ is th wavngth of th xciting ctomagntic fid and K is th cuvatu of th in aong th wi axis. Und such conditions to stabish th connction btwn potntias on th bounday of coatd and non-coatd wis within th nighbohood of th wi ρ, << λ, K, wh ρ is th distanc fom th wi in th oca coodinat systm, is th ngth aong wi axis, on can consid an ctostatic pobm fo th staight wi s Fig.. This ctostatic pobm can b fomuatd as foows. Th is a distibutd chag on th bounday of th wi with dnsity p unit squa q s q π a which cosponds to th dnsity p-unit ngth q. In th fist cas, th wi is uncoatd, in th scond cas th wi is coatd by a dictic ay with dictic pmittivity ε. Th task is to find th ε diffnc btwn th potntias on th bounday of th wi ϕ and ϕ. 4

5 Fig..: Th ctostatic pobm fo a coatd wi. This pobm can b sovd using Gauss thom in ctostatics. W consid a sction of th cyind with ngth d and som auxiiay cyindica sh with adius ρ. Accoding to Gauss thom wh q V is th voum dnsity of chag V d div dv qv dv qs ds qdz qd taking into account that V V S div dv S ds πρd wh an obvious cyindica symmty of th ctica fid distibution is usd on can obtain an quation fo th ctica dispacmnt q...3 πρ and fo th ctica fids, both, in th fist and scond cas: q E ρ, a ρ πε ρ <...4 E ε ρ q, πε ερ q, πε ρ a < ρ < b ρ > b...5 If now w choos as fnc point fo th potntia som point R b << R << λ, K w can wit fo th potntias on th bounday of th wi in both cass and fo thi diffnc: a ϕ Eρ dρ...6 R 5

6 ε a ε ρ ϕ E dρ...7 R a a ε ε q q q ε b ϕ ϕ E E dρ dρ n...8 ρ ρ πεε ρ πε ρ πε ε a R b o ϕ ε ϕ q πε ε b n ε a...9 If w now tun to th initia ctodynamic cas, w can consid th potntia ϕ as th potntia of th uncoatd wi, and us th xpssion fo th chag p-unit ngth which foows fom th continuity quation j I q ω... w can obtain th dsid connction btwn th potntias: I ϕ ε ϕ ~... jωc wh som auxiiay capacitanc is intoducd ~ πε ε C ε n b a... Now w consid a thin wi of abitay gomtic fom, wh is th coodinat of th wi s axis, na th pfcty conducting gound, which is xcitd by an xtna fid E i. Assuming, as usua in th thin-wi appoximation, that th cunt I fows aong th wi axis, and using th continuity quation... w can obtain th foowing quation fo th scaa potntia in th Lonz gaug on th bounday of th uncoatd wi: L ϕ C I g I, jω4πε d...3 H L d is haf of th ngth of th compt cosd oop. Th function g I C, is th Gn s function aong th cuvd in fo th scaa potntia, which taks into account th fction of th gound pan: g C I, jk [ ] [ ] a a jk [ ~ ] [ ~ ] a a...4 6

7 H ~ is th adius vcto of th axis miod by th gound pan, and a is th adius of th wi. Using..4 and... w can wit th dsid scond MPIE quation fo th potntia of th coatd wi L g I 4πε, d ~ C I ε 4πε jωϕ C I...5 ~ H th auxiiay capacitanc C is dfind by q.... and contains th dictic pmittivity ε. If th ossy dictic is considd, th scond tm in quation...5 is sponsib fo th dictic osss... Th fist MPIE quation: Bounday condition fo th finit conductiv wi. Again w consid a thin wi of abitay gomtic fom na th pfcty conducting gound. It is xcitd by an xtna fid E i. In th pvious considation fo th pfcty conducting wi [], w assumd that th tota initia pus scattd tangntia ctica fid on th bounday of th wi is zo, which ad to th fist MPIE quation i sc i ϕ E E E jωa tot E... i wh E and A a tangntia componnts of th xciting fids and vcto potntia spctivy. Using th xpssion fo th vcto potntia on can -wit... as ϕ L µ L i jω gi, I d E 4π... wh th function g L I, is th Gn s function fo th tangntia componnt of th vcto potntia in th Lonz gaug aong th cuvd in, which taks into account th fction of th gound pan: g L I, jk [ ] [ ] a a ~ jk [ ~ ] [ ~ ] a a...3 Th unit tangntia vcto of th cuv is takn aong th wi axis, ~ is th adius vcto miod by th gound pan, and ~ ~ is th cosponding unit tangntia vcto. If th tansmission in is not pfcty conducting, th tota tangntia ctic fid on th conducto in not zo. In this cas th bounday condition... o th quation 7

8 ... can b modifid to tak into account th psnc of a finit ctica conductivity of th wi. This is don by intoducing a sufac impdanc appoximation, which ats th tota oca ctic fid on th sufac of th wi to th tota tangntia cunt fowing on th wi at th sam point []. This ationship is xpssd as E i ϕ E jω A Z wi...4 tot H Z w jω is th p-unit-ngth impdanc of th wi[,3]: Zcw I γ wa Z w jω...5 πa I γ a w In q....5 th functions I x and I x a modifid Bss functions, Z cw is th wav impdanc in th conducting composition of th wi givn by Z cw jωµ...6 σ w and th tm γ w is th popagation constant in th wi matia, γ w jωµ σ w...7 Th ctica conductivity of th wi is dnotd by σ w. In this discussion, th wi is assumd nonmagntic, with unit pmittivity. Vaious appoximations to this wi impdanc a possib [4]. At ow fquncis, wh γ wa <<, this impdanc is givn by Z j ω µ jω RC π w πa σ w 8 jωl in...8 and at high fquncis, wh γ wa >>, this impdanc bcoms j µ ω Z ω...9 w j πa σ w With th intoduction of th wi impdanc in th xpssion fo th tangntia ctica fid...4 and appying th sam stps as don pviousy, th foowing fist MPIE fo th ossy wi suts: ϕ L µ L i jω g I, I d Z w I E 4π... Ths two MPIE... and...5 a th main suts of th two fist sub-sctions. 8

9 It is possib to show, that fo th patia cas a staight finit wi in th f spac, th scond od Pockington-ik intgo-diffntia quation fo th inducd cunt, which can b divd fom... and...5 fo th cas Z w, coincids with suts of paps [5], [6] fo th coatd thin staight wi.. Fu-Wav Tansmission Lin FWTL quations fo th wiing systm with osss. Itation appoach fo th goba paamts. Th two MPIE obtaind in th pvious sub-sctions a a stating point fo th divation of th goba paamts in th Fu-Wav Tansmission Lin Thoy FWLT. In this Sction w shoty outin th divation, fing th ad to [],[7],[8]. Lt us consid again th systm of MPIE fo th thin coatd, ossy wi of abitay gomtic fom.. a,b L ϕ µ L i jω g I, I d Z wi E 4π L C I 4πε I g I, d 4 ~ πε jωϕ C.. a,b In od to dfin th goba gnaizd tansmission in paamts, w consid an xcitation of th tansmission in by point soucs with abitay dimnsionss ampituds U and U ocatd at th bginning and at th nd of th in, which cosponds to th tangntia componnt of th xciting ctica fid: E i U δ U δ L V with.. It is possib to show, that th xcitation fid fo th oadd in fomay can b ducd to th sam quation [,7,8]. Fo xamp, whn th systm is oadd by th impdanc Z at th ight tmina, thn th cosponding constant is U I L Z / V..3 Lt now th functions I, I and ϕ, ϕ b soutions of th systm.. a, b fo th cunt and th potntia with soucs with of ampitud V: δ, δ L ocatd in th points and L, cospondingy. u to th inaity of th considd pobm w can wit th soution fo th tota inducd cunt as I U I U I..4 Fo th potntia ϕ aong th wi w find a simia quation: ϕ U ϕ U ϕ..5 9

10 Now w a ady to ook fo th systm of diffntia quations Fu Wav Tansmission Lin quations, FWTL fo th potntia and cunt outsid th souc gion in th Tansmission - Lin - Thoy - ik fom. dϕ jωp d di jωp d I ϕ jωp jωp ϕ I..6 To do that, w bgin to us a matix notation this way of soution can b gnaizd fo th muticonducto cas. W intoduc a coumn-vcto x, which componnts a potntia and cunt, as fo th patia, as w as fo th gna soution : x : ϕ ; I x : ϕ ; I x : ϕ I..7 Now th quations can b wittn in th foowing fom: U x U x x..8 and, instad of..6 w hav: d d x ω [ P] x, wh [ P j, ] j P jω,, P jω, ω,..9 a,b P jω,, P jω, Eqs...9 a and..8 man that U d d ω.. [ P] x U x jω [ P] x x j d d sinc th U and U hav abitay vaus in.. w div d d d d x [ P] x j x ω.. a [ P] x j ω.. b Aft an intoduction of a matix notation fo th patia soutions ϕ I [ X ]: x, x ϕ I.. q... a,b can b wittn as

11 d d [ X ] j [ P] [ X ] ω..3 If th matix of patia soutions x is non-dgnatd, i.. [ ] ϕ I ϕ I : dt X I, ϕ..4 than th soution fo th matix [ P ] can b wittn in th foowing fom [ P ] [ X ] [ ] d X jω d..5 In th usuay usd scaa notations..5 can b wittn as P P P P ϕ I I ϕ..6 a j ω I, ϕ ϕ ϕ ϕ ϕ..6 b j I I j..6 c ω I, ϕ I ω I,ϕ ω I, ϕ I I ϕ ϕ I..6 j It is asy to show, that if w do not stat th cacuation fom th functions I, I, ~ ~ ϕ, ϕ, but fom som non-dgnatd ina combinations of thm: I, I, ~ ϕ, ~ ϕ o X ~ in th matix fom with [ ] [ X ] [ α ] X ~, wh dt[ ] th sut fo th matix [ P ] wi b kpt th sam [ ~ d ] [ X ] [ α ] P jω d X jω d d [ ] [ ] X [ P ] α..7 [ X ] [ α ] [ X ] [ ] [ ] [ ] α α d jω d X..8 Thus, w hav shown that th systm of MPIE.., th soution of which is dfind by two indpndnt constants can b xpicity ducd to th diffntia quations..6,..9 a with paamts..5,..6. Ths paamts goba paamts in th Fu-Wav

12 Tansmission Lin Thoy o th paamts of Maxwian cicuits a compx vaud, and dscib th adiation of th systm []. Thy dpnd on th gomty of th systm, and thfo on th oca paamt aong th in. This fact was stabishd ai in [8-] with th mthod of th poduct intga, and up to notation in [9] by pocssing th numica soution fo th cunt and potntia with th Mthod of Momnts. Th soution of th systm..6 with paamt matix P..6 and usuay usd bounday conditions fo th cunts and votags diffncs of potntias fo th sma in th points and L yids th cunt and votag distibutions aong th in fo abitay givn vaus of th tmina soucs and/o oads. Th pocdu is convnint, whn th xact vaus of th functions I, I, ϕ, ϕ a known, fom anaytica [,8] o numica [9] soutions. Anoth way to obtain th matix of goba paamts apioiy is to oganiz som itation pocdu fo this matix. Gnay, at th zo stps, th appoximat soution of th systm..6 is dfind. Thn this soution is usd to find th cosponding paamts, tc. In [8-] as zo itation th static distibutions fo th cunt and potntia a usd, and th fist itation fo th paamts was obtaind aft som numica pocdu. Anoth pocdu, which is basd on th thicknss of th wi, was poposd in [], wh, at th zo stp th MPIE.. within th ogaithmica accuacy was ducd to th cassica TL systm with constant paamts. Th soution of this systm with soucs.. yids th cunt of th fist itation, I and I, th ina combination of which up to th constant facto can b psntd as fowad and backwad popagating wavs I xp jk I xp jk..9 Howv, fo th scaa potntia in th fist itation and fo its divativ, th xact quations.. a,b a usd. Aft som staightfowad cacuations, w obtain th goba paamt matix in th fist od appoximation: L L P c C C ~.. a C Z w Z w L L j C C j C C P ~ ~ ω ω C C ~.. b C P C C ~.. c C

13 3 C C C C C c P ~.. d In qs... w hav usd th foowing xpssions fo th fowad and backwad inductanc and capacitanc of th fist od, spctivy: d jk g L L L ± xp, 4 m π µ.. d jk g C L C ± xp, 4 m πε.. Fo th ow-fquncy cas k w find : L L L [ ] [ ] d a a L ~ ~ 4 π µ..3 : C C C [ ] [ ] d a a L ~ 4 πε..4 Th quantitis L and C constitut th a, ow-fquncy ngth dpndnt inductanc and capacitanc p unit ngth fo th ossss uncoatd tansmission in [9]. Thn, using.. fo ou cas w obtain th paamt matix in th cassica anti-diagona fom fo th coatd wi with osss. [ ] ε ω C j Z L P w k..5 wh w hav intoducd th p-unit ngth capacitanc fo th coatd wi C C C ~ ε..6

14 3. Popagation of Cunt Wavs aong Quasi-Piodica Wiing Stuctus: a Quantum Mchanica Anaogy. 3. FWTL and Schöding-ik quations W consid a ong ossss thin conducto of abitay gomtic fom abov a pfcty conducting gound, wh th non-unifomitis a ocatd in th cnta pat of th conducto. W assum, that th soucs a ocatd at th ft nd of th wi at minus infinity and that th fction is absnt fom th ight nd of th wi pus infinity. As was shown in th Sction, th cunt and potntia aong th in a dscibd by th FWTL..6 with th matix of goba paamts [ P ]. In ou considation th Lonz gaug fo th potntia is usd, but, of cous, w can us any anoth gaug, fo xamp, th Couomb gaug. Howv, fo th cunt w obtain a gaug indpndnt diffntia quation of scond od [9,]: I U M I TM I 3.. H is th ngth-paamt of th cuv takn aong th wi axis, U M is th compx damping function, T M cosponds to th squa of th popagation constant. Ths paamts a connctd with th goba paamts of th MPIE [] and thy aso dpnd on fquncy and on th gomty of th systm. d U M n P jω P P d 3.. d P T M jωp ω dt[ P] d P 3..3 To duc q. 3.. to th fom convnint fo futh anaysis, w iminat th fist divativ by intoduction of a nw unknown function ψ : I f ψ 3..4 f xp U d P xp P jω P P M 3..5 Th function ψ satisfis th diffntia quation of scond od 3..6 ψ k ψ ; 3..6 k du M U M : TM 3..7 d 4 To consid th wiing stuctu at th unifom nds w intoduc a potntia function u as foows 4

15 ω k k > im ± c 3..8 u k k 3..9 u im ± 3.. k u ψ ψ 3.. Eq. 3.. ooks ik a Schöding quation in non-ativistic quantum mchanics with th potntia u [5]. As w as th paamts T M and U M th potntia u dpnds on th gomty of th wi and on th fquncy. Fo ow fquncis, kh wh h is th hight of th wi at ±, using and on can show, that th potntia can xpssd as u ω d C ε 3 dc ε ω L Z w jω C ε c C ε d 4 C d ε 3.. Fom 3.. it is obvious that fo th ossss in th potntia is a and q. 3.. dscibs th on-dimnsiona scatting of a quantum mchanica patic, whn th numb of patics is kpt constant [5]. In th gna fquncy cas, whn w hav to us in 3.., 3..3, 3..7 th suts of qs..., th potntia bcoms: u k d jω d d n [ ] P P d P P jω P P n P jωp d d P P 4 ω dt d d 3..3 Now as w as fo th ow fquncy cas with ohmic o dictic osss th potntia is a compx-vaud quantity that cosponds to osss in th initia ctodynamics pobm and cosponds to absoption of th patics in th quantum mchanica anaogy. 3. On-dimnsiona scatting pobm. Rfction and tansmission cofficints fo th ossy systms. In th pvious sub-sction w hav shown, that th homognous ctodynamica pobm is quivant to th on-dimnsiona quantum mchanica scatting pobm, wh th patic coms fom minus infinity, scatts at th potntia, bcoms patiay absobd in th potntia gion, popagats patiay though th potntia and aso is patiay fctd by th potntia. Th compx quantum mchanica ampituds of ths pocsss a dscibd by th compx fction R and tansmission cofficints s Fig. 3. 5

16 Fig. 3.: On-dimnsiona quantum-mchanica scatting pobm xp jk R xp jk fo ψ 3.. xp jk fo Ths cofficints bcom vy impotant in this pot and w now dscib thi poptis in mo dtai. Fo symmtica scatting potntia, u u, ths cofficints a th sam fo th ft and fo th ight scatting pobm. It is possib to show that fo th ow-fquncy ossss cas wh th potntia u is a ths cofficints satisfy th foowing quations [4]: R 3.. R{ R * } 3..3 Howv, fo th compx potntia cosponding to adiation and o ohmic and dictic osss th quations a not vaid. Th imaginay pat of th potntia now is sponsib fo th osss. Lt us stabish this dpndnc. If w hav a unifom in with cunt wavs popagating in both dictions and th fid aound wi is a TEM wav I I xp jk I xp jk 3..4 th tim avagd pow popagating aong th unifom in in th positiv diction can b wittn as I I ZC W 3..5 Eq can b psntd as * Z c I * I W I I 4 jk

17 Lt us now dfin th vau W by q not ony fo th two asymptotic gions ± but aso fo th intmdiat gion gion of intaction. Now w can xpss th ngy osss of th cunt duing th scatting pocss causd by adiation, ohmic o dictic osss using th aw of ngy consvation W W oss W 3..7 Using th psntation of th cunt though th ψ -function 3..4 w can wit fo th quantity W : * ZC ψ * ψ W xp R U M d ψ ψ Im U M ψ 4 jk 3..8 Fo a symmtica wiing systm, fo xamp fo th wi with vtica coodinat x and hoizonta coodinat z, which a givn by th ations v x,, z ; x,, z 3..9 a,b W find aft som combsom cacuation with th aid of th tchniqu fom Sction, th foowing symmty poptis fo th goba FWTL paamts: P P 3.. a,b,c,d P P P P P P Now, using th dfinition of th paamt U M 3.. and q. 3.. w can find that U U, U M d 3.. a,b M M Fo q w thn obtain W ± and, consqunty W oss * ZC ψ * ψ ψ ψ jk R ZC I

18 8 Fo th ossss cas w obtain th obvious answ: oss W. W wi mak som standad manipuation with th Schöding quation 3.. fo th ossy cas to obtain th tm in th backt Lt us consid th Schöding quation 3.. fo th function ψ and fo th compx conjugat function * ψ. Mutipy thm, cospondingy, by * ψ and by ψ, thn subtact th scond fom th fist quation: - * * * * u k d d u k d d ψ ψ ψ ψ ψ ψ 3..4 a,b As sut, w hav Im * * * * * * * u j d d d d d d u u d d d d ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ 3..5 Intgating 3..5 fom to and using 3.. yids: * * Im R jk d u j d d d d ψ ψ ψ ψ ψ 3..6 Taking into account 3..6 and 3..3, w finay hav d u k I Z R I Z W C C oss Im ψ 3..7 Th quation 3..7 givs th dsid connction btwn th imaginay pat of th potntia and th osss in th wi. 3.3 Tansf matics fo th on-dimnsiona scatting pobm. In this sub-sction, w consid anoth, mo gna appoach to dscib ondimnsiona scatting, which is givn by th mthod of tansf matix [6]. In this mthod, w consid wavs popagating in positiv and ngativ dictions with diffnt ampituds, both fom th ft and ight sids of th potntia 3.3. s Fig. 3.

19 C xp jk C xp jk fo ψ 3.3. ~ ~ C xp jk C xp jk fo Fig. 3.: On th dfinition of th tansf matix S. Bcaus th considd pobm is ina, th xists a ina connction btwn th asymptotic ampituds of th potntia. This connction is aizd by tansf matix S S S S S 3.3. which inks th coumn vctos of th wav ampituds: ~ ~ C C C : ~ S : S C C C Lt us now xpss th tansf matix in tms of fction and tansmission cofficints R and, spctivy. Fo th ft sid scatting pobm Fig. 3., hav th foowing componnts: C : and R ~ C : a,b Simiay, fo th ight hand sid scatting pobm, whn th patic incidnts fom pus infinity, w hav C : and ~ R C : a,b Substitut into 3.3.3, and find two componnts of th matix S : R S and S a,b 9

20 Equations fo th two oth componnts of th tansf matix a obtaind aft substitution of th into S S S S R o, S S S S R S R S R and yid th dsid quation fo th matix S R a,b R S R R On can asiy chck that dt S 3.3. Fo a a potntia which cosponds to a ossss systm w us th poptis of th fction and tansmission cofficint to show, that [4,5] jϕ R n j * S 3.3. * R jϕ n * j jϕ wh, and th numbs n,,, Kcospond to th numb of boundd ngy stats in th potntia 3... Aso, fo th ossss systm w hav S S * 3.3. Th poptis of th tansf matix fo th ossss cas can b wittn in diffnt ways: * S ; S ; dt S S S * S S a,b,c Thfo, th tansf matix S in this cas bongs to th SU, goup of matics [6,]. uing mutipication this matix kps th vau

21 ~ ~ C C C C ~ W const constant, which physicay cosponds to th patic fux pow consvation aong th in. Fo th f patic cunt wav popagation without any scatting, w aso can intoduc th tansf matix. In this cas th tansation ducs to th chang of th coodinat. To s this w consid th f popagation wavs fo two oigins of coodinats and s Fig. 3.3: Fig. 3.3: F popagation of th cunt wav C jk C jk jk C 443 ~ C jk jk C 443 ~ C jk Fom th th tansf matix fo th f popagation is divd as: ~ C ~ C C C jk jk : T C C, Of cous, th tansf matix fo th f popagation aso bongs to th SU, goup of matics. 3.4 Tansf matix fomaism fo th quasi-piodica systm with osss. Lt us now assum that th quasi-piodica wiing stuctu is fomd by a finit numb of idntica sub-mnts s Sction 4. W assum that th cosponding potntia in quation 3.. can b psntd as a st of piodicay aangd idntica potntias : W now consid th popagation though a chain, consisting of N potntias, spaatd by asymptotic gions, wh th potntia is appoximaty zo s Fig Lt us assum that th coumn vcto on th ight sid of th quasi-piodica systm w assum now, that th bginning of th coodinats is in th cnt of th fist potntia is C. Aft th fist scattd th coumn vcto bcoms S C. Changing th coodinat oigin by It is possib to show, that this assumption btt satisfis fo th cas of ow fquncis, but aso fo high fquncis, whn adiation dos not dominat, this assumption is aso appoximaty vaid. Moov, th mo sub-mnts a considd th btt satisfis this assumption.

22 th f popagation tansf matix T : T, to th ight sid of th scond potntia, w hav a coumn-vcto S C. Rpating this pocss up to th ast potntia, w can T wit fo th tota tansf matix, with th bginning of coodinats in th cnt of th ast potntia Fig. 3.4: Popagation though th quasi-piodica chain of potntias N 5. T t N, N : T N, N T n, n n S n 3.4. Aft th tun to th bginning of th coodinats to th cnt of th fist potntia w hav fo th cosponding tansf matix SΣ S 3.4. N N T, N T n, n n n Fo th cas of piodicay aangd idntica potntias q can b wittn in th foowing fom S T, NL T L, S N Σ N Now, it is mo convnint to mov th coodinat oigin in th cnt of th quasipiodica systm. This can b don if th cosponding tota potntia is symmtica, and Th matix 3.4. is connctd with an xpicit fom of th so-cad poduct intga [9] which yids th soution of th scond od diffntia quation 3.. fo th cas whn th gions of intaction a dividd by th gions of zo potntia. W mntion h that th st of tansf matics of th fom 3.4. fom a goup: t t t Th is dfind a poduct opation T 3, T, T 3, ; Th xists a unit mnt T t, I ; Each mnt has invs mnt T t T t t t ; T T I,,,,

23 ft and ight fction and tansmission cofficints a th sam. In this cas, q can b -wittn as N SΣ T, N L / T L, S T N L /, N o, using 3.3. and S Σ N Σ N R Σ N R Σ N Σ N Σ N R Σ N Σ N Σ N Th knowdg of th tota tansf matix givs us th possibiity to obtain an quation fo th tota tansmission cofficint in xpicit fom, xpssd by th scatting data fo on potntia. To cacuat th tota tansf matix, w woud ik to obtain a simp anaytica th xpssion in th N pow in q Fo this pupos, it is nough to find an xponntia psntation of th matix T L, S with som additiv paamt 3. Lt us bgin with th cas of a ossss potntia [4]. In this cas, on can obsv that th matix T L, S, as w as both of th factos bong to th SU, goup. Now, on may mmb that fo th matix of finit otation fo th spin of patics with spin /, which bongs to th SU goup U U U ; dt U * * ; U U ; U U U U th xponntia psntation bcoms [5] 4 : U xp jϕ nσ / I cos ϕ / jnσ sin ϕ / wh ϕ is a a additiv paamt th ang of otation aound th unit vcto n n has a componnts and n. σ is on vcto of th Paui matics, which fom togth with th unit matix I, a basis fo matics I σ ; x j σ ; y j σ ; z 3 Th paamt χ, which is th agumnt of th goup mnt g χ, is cad an additiv paamt, if fo any two mnts of this goup χ g χ g χ χ 4 Th scond quaity in fo th unit vcto n can b obtaind, using th anti-commutation poptis of th Paui matics: σ i σ j σ jσ i δ and th dfinition of th matix xponnt fom th sis. i, j 3 g.

24 Th poptis of th goup SU a simia to th poptis of th goup SU,, with th xcption that a SU tansfomation kps th foowing vau constant: ~ ~ C C C C const, instad of Thfo, w ook fo an xponntia psntation of th matix T L, S with diffnt vaus of th paamts. In [4] it is shown that this appoach is succssfu, howv, th ang of otation ϕ, as w as som componnts of th unit vcto n can b compxvaud. Now w appy th sam appoach to th gna cas of a tansf matix fo th ossy systm not th SU, cas!. W ty to ook fo th psntation of th matix T L, S in th fom 3.4.6, namy T L, S xp 3.4. jϕ nσ I cos ϕ jnσ sin ϕ in th q w us th paamt ϕ instad ϕ / fo convninc. Now w ty to xtact th paamt ϕ and n x, n y, nz fom th q Th ft sid of 3.4. is T T L, S : T t t L, L, T T t t L, L, R R R 3.4. Th ight sid of 3.4. can b wittn as xp I cos j ϕ nσ I cos ϕ jnσ sin ϕ ϕ jn σ sin ϕ jn σ sin ϕ jn σ sin ϕ x x y y z z 3.4. cosϕ jnz sinϕ jnx ny sinϕ jnx ny sinϕ cosϕ jn sinϕ z W can find paamts of th xponntia psntation by compaison of 3.4. and Th additiv paamt ϕ can b found by taking th sum of th diagona mnts: t t L, T L, R : α cosϕ T Th coodinats of th vcto n x and n y can b found by summation and subtaction of th non-diagona mnts, cospondingy: 4

25 j n x t t R T L, T L, j sin kl sinϕ o n x R sin kl sinϕ n y t t R T L, T L, cos kl sinϕ o R cos kl n y sinϕ Th z -componnt of th vcto n can b found by subtaction of th diagona mnts: j n z sinϕ t t T L, T L, R n z R j sinϕ W not that diffnt fom th ossss cas, wh th paamt ϕ is a o pu imaginay, in th ossy cas this paamt is compx in any cas s aso th Sction 3.5. As in th cas of a ossss potntia [4], th vcto n now is compx vaud 5 and it is possib to show it squad vau bcoms on: n Now, having th matix T L, S in th xponntia fom with additiv paamt ϕ, th w can asiy wit fo th N pow of it: cos Nϕ jn sin Nϕ jnx n y sin Nϕ jn n sin Nϕ cos Nϕ jn Nϕ z sin N N z x y T L, S { xp jϕ nσ } xp jnϕ nσ sin Nϕ cos Nϕ β sinϕ R sin Nϕ sinϕ R sin Nϕ sinϕ sin Nϕ cos Nϕ β sinϕ w w hav intoducd th notation t t T L, T L, R β : And, aft simp matics mutipications, w hav fo th tansf matix S T, N L / N T L, S T N /, Σ N L 5 Th compxity of th componnts of th unit vcto n, in contast to th a unit vcto fo th opation of finit otation, is causd by th fact that th tansf matix T t L, now is not hmitian. 5

26 N N cos Nϕ jn sin Nϕ jn x n y z sin Nϕ sin Nϕ cos Nϕ β sinϕ R sin Nϕ sinϕ N N jn x n y sin Nϕ cos Nϕ jn sin Nϕ R sin Nϕ sinϕ sin Nϕ cos Nϕ β sinϕ z 3.4. Compaing with th quation fo th tota tansf matix, xpssd though th tota fction and tansmission cofficints 3.4.5, w can find fo ths cofficints: R N ; Σ N R sin Nϕ sinϕ sin Nϕ cos Nϕ β sinϕ N N 3.4. a,b Σ sin Nϕ cos Nϕ β sinϕ Eq can b wittn in anoth fom using th dfinition of Chbyshv poynomias of th fist, T N x, and scond, U N x, kind []. N accos α T α cos N ϕ cos 3.4 a,b N N accos α α U a sinϕu α sin N ϕ sin N N R U N α R Σ ; N T α β U α N N N N Σ ; a,b N T α β U α N N Using th dfinition of th auxiiay vaus α and β w can wit th quations fo th tota fction and tansmission cofficints of th piodica chain of ondimnsion scatts in th xpicit fom: R Σ N Σ N R T T N N R R N U N R R U N R N U R N R Th quations a vy impotant suts of th psnt pot. 6

27 W not, that th q. fo th N th pow of th matix T t L, and a fomua foowing fo th fction and tansmission cofficint can b aso obtaind if on uss th standad mthods to diagonaiz matics [], fo matics with unit dtminant. Th psnt way, howv, is ca and can b a basis fo futu gnaization in th cas of muticonducto wis. Lt us now shoty invstigat som spcia cass of qs Fo th cas of a a potntia ossss in, using qs. 3.. and 3..3, w can wit α cosϕ R ; β Im a,b and, haft, a ducd to th sut of pvious pot [4]: R Σ N R T N N U N R R j Im U N R N Σ N T N R j Im U N R Fo th cas of on scatt N potntia of gna viw, w us th vaus of Chbyshv poynomias T x x, U x and find fom q an obvious sut Σ ; R Σ R ; a,b Fo th cas of two scatts N using th vaus of th Chbyshv poynomias T x x, U x x it is possib to find ; Σ R R R R ; a,b Σ R which can aso b obtaind by th appication of Fynman diagams fo th on dimnsiona scatting 6. 6 uing this cacuation w us th facts that th cofficints R and a quantum-mchanica compx ampituds of th fction and tansmission vnts of on-cnt scatting. Th compx ampitud of th f popagation btwn points and is givn by th xponnt function xp jk. Aft that on can consid diffnt quantum mchanica pocsss, which ad to th popagation though o fction fom th chain of two potntias pntation though th fist potntia, popagation btwn potntias, popagation 7

28 If th potntia, which cosponds to on spaat non-unifomity, is not asy to pntat, R, << on can obsv a sonanc scatting. To mak a shot quaity invstigation of this phnomna w consid th q a and not that th on-potntia fction and tansmission cofficints and R hav mo sowy fquncy dpndnc in compaison with th xponntia function xp. Lt us invstigat now th fquncy dpndnc of th tansmission cofficint Σ. Fo th main fquncy gion, whn kl ϕ R πn ~ wh ϕ R is a phas of th tansmission cofficint R, n,,3.. th dnominato in has an od of magnitud on, and, by this way, th pntation though th two-potntia chain is sma Σ ~ <<. Howv, in th naow fquncy bands, whn kl ϕ R πn <<, on can obsv a sonant scatting. By th intoduction of th dtuning kn kl ϕ R π n of th n th sonanc on can wit fo th fquncy dpndnc of th tansmission cofficint in th nighbohood of this sonanc th nxt quation: Σ R j k R n Th q dscibs a typica sonanc fquncy cuv. Fo th zo vau of th fquncy tuning a vau of th popagation cofficint stongy incass Σ s R Fo th cas of th ossss potntia Σ s and it absout vau is on. On can show [3] that in th cas of sonanc scatting th patic is jammd btwn ths potntia pits bais, and has mutip -fctions. It spnds a ong tim insid th chain of potntias that ads to th incas of th wav function ampitud insid th chain. In th ctodynamics anguag, duing th sonant scatting, th cunt ampitud btwn two scatts incass. If w da with ossy systms, th osss of any natu ohmic, adiation, tc. hav to incas. This, fo xamp, can b impotant fo th intnsity of adiation of th systm and fo th sistanc of th systm with spct to ohmic hating. though th scond potntia, fction fom th fist and scond potntia and us th quantum mchanica axioms, which stat:. Th ampitud of two indpndnt vnts is th sum of thi ampituds;. Th ampitud of two vnts in squnc is a poduct of thi sing ampituds. 8

29 3.5 Aowd and fobiddn fquncy zons. Connction of th paamts of quasipiodica and piodica systms. In th psnt Sction, w us th obtaind suts to invstigat th tansmission cofficints and to stabish a connction btwn paamts of quasi-piodica and piodica systms. Fist, w consid th ossss cas. Fo such systms th quation of th otation ang ϕ main banch can b wittn a fo diffnt magnituds of th vau α as R, R, R, j accosh R ϕ accos R 3.5. π j accosh R If th wav numb fquncy is such, that th paamt cosϕ R, w hav in th dnominato of 3.4. b osciating functions, and th tota popagation cofficint is of th od of magnitud on. In th opposit imiting cass R o R, w hav th hypboica function in th dnominato and th popagation cofficint is xponntiay dampd Σ N ~ xp Nϕ fo N >>, and th fction cofficint is about on. Thus, w hav shown that aowd and fobiddn zons appa fo th finit chain of potntias. Ths fquncy zons a cad aowd and fobiddn, cospondingy, bcaus in th aowd zon fo N th patic can pntat insid th smi-infinit chain and cannot b in th fobiddn zon. Th xistnc of th aowd and fobiddn zons is w known in soid-stat physics fo an infinit piodica potntia [7]. Th good pntation though th finit chain of potntias fo th aowd zon can b physicay xpaind as a sonanc scatting on th quasi-stationay ngy vs of th chain of potntias, which appa bcaus of th spitting of th quasi-stationay ngy vs in th systm of two potntias pits bais s th nd of th pvious Sction. In this cas, again on has R, <<, th patic is jammd btwn ths potntia pits bais and has mutip -fctions. Again, th wav function cunt ampitud stongy incass, but fo th chain th incas can b much stong in compaison with th cas of two potntias. This phnomna, numicay was invstigatd in [4] and can sv as basis to constuct adiating dvics, as w as to invstigat th sistanc of a piodica systm with spct to ohmic hating. It is possib to show that th incusion of adiation osss ducs th pntation in th aowd zons, s Sction 4 of th psnt pot, but th stuctu of th aowd and fobiddn zons mains vaid. Now w stabish th connction btwn th paamts of th popagation of th patic though th infinit chain and invstigat paamts of on-cnt scatting. If th patic popagats though th infinit chain of potntias with piod L, u L u L 3.5. It s wav function can b psntd as [7] Foqut thom: 9

30 ψ xp jk Ψ, Ψ L Ψ wh th function Ψ is piodica with piod L, and th paamt K, cad quasipus mo xacty quasi-wav numb, chaactizs th tansation poptis of th wav function it may b positiv as w as ngativ. ψ L xp jk L Ψ L xp jkl ψ Rmmb that in ou cas of piodica potntia pits bais, spaatd by asymptotic gions, th wav function can b psntd in th asymptotic gion as ψ C xp jk C xp jk Th tansf matix T t L, fo th coumn vcto C fo on piod tansation is givn by q On th oth hand, q yids th foowing xpssion fo th on-piod tansf matix T t L, xp jkl I Equaizing th tansf matics 3.4. and ads to a homognous ina systm fo th coumn vctos: R R, R C C jkl C I C which is sovab, if dt R R, jkl R jkl This quation yids as a sut fo th quasi-pus cos R KL cos ϕ 3.5. Th quation 3.5. stabishs th connction btwn th quasi-pus quasi-wav numb K and th usua wav numb k ω / c. It is th so-cad dispsion quation [7]. Fom th 3.5. and 3.5. on can obsv, that fo th aowd zons th quasipus is a fo th ossss potntia and th patic can popagat aong th infinit chain. 3

31 Fo th fobiddn zons th quasi pus is imaginay and th popagation disappas. In cas of a ossy potntia th imaginay pat of th popagation constant infuncs th patica popagation though th aowd zons by som dcmnt. Now w want to add a fw wods about th popagation of ngy in th infinit piodica systm. Using th fomua fom Sction 3., on can obtain th foowing quation in th asymptotic gions fo th avagd pow popagating aong th in W C C ZC I 3.5. Haft, it is ncssay to us th connction of th cofficints C and C fom q Omitting quit awkwad invstigations, w fomuat h th finit suts: fo som vn aowd zons th diction of th popagation of th phas th sign of th quasi-pus K coincids with th diction of th popagation of th ngy hav th sign of Fo oth odd aowd zons, ths dictions a opposit. It sms to b that this fact is connctd with th xpimntay stabishd [4] connction of th phas and goup vocity of piodicay oadd tansmission ins, whn, fo th som fquncy bands, thy hav opposit dictions. 3

32 4. Numica xamp. Rativ impotanc of diffnt ossy mchanisms. In this Sction, w consid a spcific xamp and appy th dvopd mthod. W consid a wi of adius a cm, which cam fom minus infinity at a hight of h m, xpincs sva osciations, and uns to pus infinity. Th non-homognous pat of th wi consists of 5 idntica sctions of a Gaussian fom 4. s aso Fig. 4.. n k z nl x z h b xp xz, m z, m Fig. 4.: Gomty of th quasi-piodica wiing stuctu: h m, b.75 m, k. 75m, a. m, L 8 m, N 5. Fist w consid a pfcty conducting uncoatd wi, wh ony adiation osss a possib. Th mnts of th matix of th cosponding goba paamts [ P k, ] of th Fu-Wav Tansmission Lin can b cacuatd using of ou ptubation thoy qs Ths paamts a compx-vaud and coodinat-dpndnt s Figs. 4. a,b,c,d. To chck ou cacuation, w compa th suts fo k with th static suts fo P k, L and P k, C obtaind by q s Fig:4.3 a,b. k k k k Sinc th considd wi systm is symmtica aound th oigin of coodinats, on can obsv that th diagona paamts a symmtica and th anti-diagona paamts a anti-symmtica aound th oigin of th coodinats s q Haft w cacuat Mi s paamts U M and T M in th scond od diffntia quations s Fig. 4.4 a,b 3... W not that fo th symmtica wi U M is an antisymmtica function of U M U M and th intga confim th asoning at th nd of th sc. 3. q U M d, which 3

33 Gauss chain, N5,h m, a. m, b.95 cm, L8 m, 4.x - k m - 3.x -.x - RP jω, ImP jω, P jω,, c/m.x -. -.x - -.x - -3.x - -4.x , m - Fig. 4. a P jω,, H/m.x -6.x -6 9.x -7 8.x -7 7.x -7 6.x -7 5.x -7 4.x -7 3.x -7.x -7.x -7. RP jω, ImP jω, Gauss chain, N5,h m, a cm, b.95 m, L8 m, k m - -.x , m Fig. 4. b 33

34 P jω,, F/m.4x -.x -.x -.8x -.6x -.4x -.x -.x - 8.x - 6.x - 4.x -.x -. RP jω, ImP jω, Gauss chain, N5,h m, a cm, b.95 cm, L8 m, k m , m Fig. 4. c Gauss chain, N5,h m, a cm, b.95 cm, L8 m, k m -.x - P jω,, c/m.x -. -.x - -.x - RP jω, ImP jω, , m Fig. 4. d Fig. 4.: Matix of th goba paamts fo th Gaussian-chain wiing stuctu: a - P jω,, b - P jω,, c - P jω,, d - P jω,. 34

35 .x -6 Gauss chain, N5,h m, a cm, b.95 cm, L8 m Inductiv cofficints, H/m.x -6 9.x -7 8.x -7 7.x -7 6.x -7 5.x -7 4.x -7 3.x -7.x -7.x -7 RP k, L' numb points of intgation z, m Fig. 4.3 a.4x - Gauss chain, N5,h m, a cm, b.95 cm, L8 m Capasitanc cofficints, F/m.x -.x -.8x -.6x -.4x -.x -.x - 8.x - 6.x - 4.x -.x - RP k, C' numb points of intgation , m Fig. 4.3 b Fig. 4.3: Compaison of th inductiv and capacitiv cofficints fo k with static inductanc and capacitanc fo th quasi-piodica wi stuctu: a - P k, and L, b - P k, and C. 35

36 Gauss chain, N5,h m, a. m, b.95 cm, L8 m, k m -..5 RU M ImU M U M, m , m Fig. 4.4 a..8 T M /k.6.4. RT M /k ImT M /k Gauss chain, N5,h m, a. m, b.95 cm, L8 m, k m , m Fig. 4.4 b Fig.4.4: Mi s goba paamts fo th piodica wiing stuctu: a - U M jω,, b - T M jω, / k. 36

37 Th tota potntia u k, fo th quasi-piodica wiing systm, which is cacuatd with th hp of goba paamts [ P k, ] is psntd in Fig Not, that th imaginay pat of th potntia is not xact piodica. In addition, it is impotant to not, that th a pat of th potntia pacticay dos not dpnd on th fquncy at ast fo k. 5 m - s Fig. 4.6 a, but th imaginay pat, which dfins adiation, is fquncy dpndnt s Fig. 4.6 b. This fquncy dpndnc can b oughy appoximatd by a quadatic fquncy function u k, ~ k. This appoximation fo diffnt points cnta maximum point of th a pat of th potntia and a ativ minimum point of th a pat of th potntia is psntd in Fig It is intsting that th imaginay pat of potntia is positiv in on point and is ngativ in anoth. Th xpanation, in ou opinion, is that this compx wiing stuctu in som points adiats ngy, but in som oth points absobs ngy. Now, with th knowdg of th potntia w can obtain th tota tansf matix of th systm and, aft that obtain a tansmission cofficint of th cunt wav though th systm. To do that, simp numica mthods hav bn dvopd s Appndix. To chck ou cacuation, w us th w-known MOM cod CONCEPT. Th configuation of th wi stuctu, which was usd to mod th infinit quasi-piodica stuctu, is shown in Fig Th infinit quasi-piodica systm was finishd at som distanc fom th piodica pat and was suppmntd by th vtica iss oadd by th chaactistic impdancs of th in Z C η / π n h / a 37.7 Ω. Th systm is xcitd by th unit votag souc U V at th ft tmina. Und such condition it is possib to show, that th votag on th ight oad is connctd with th vau of th tansf function. Of cous, this connction is vaid, if th Tansmission Lin appoximation can b appid to th asymptotica and na tmina gions of th in. In th Figu 4. th a impotant suts of th psnt pot. Th back cuv psnts th sut of th CONCEPT cod cacuation fo th configuation of Fig. 4.8, whn th doubd votag on th matchd oad with unit votag xcitation is appoximaty th tansmission cofficint of th cunt wav though th systm. On this cuv, w can cogniz th aowd and fobiddn zons. Howv, in contast with suts of pvious moding with a a potntia d cuv [4] w can s th dcmnt of th tansmission cofficint, which is causd by adiation. Th gn cuv is th sut of cacuation with th potntia fom Fig. 4.5 by th matix mthod. A quit good agmnt with CONCEPT is obsvd bfo th fouth aowd zon. Howv, bcaus w hav to cacuat th potntia in ach fquncy point diffnt fom th ow-fquncy cas th cacuation tim is vy ong and w hav to us a quit ough division of th intva. W dividd th 5 m distanc intva into subintvas and usd fquncy points. Moov, fo this ough appoximation th cacuation tim was about 3 hous!. Th diffnc of th two mthods can b xpaind by th stong adiation na th vtica mnts. On th oth hand w usd ou anaytica fomua fo th popagation cofficint of th chain. Fist, w dfin by th matix mthod th fction and tansmission cofficints s Fig. 4. though on patia cnta potntia of th Gaussian chain s Fig This cacuation is 5 tims fast than th tansf matix cacuations fo th tota systm. Aft that, w usd ou anaytica fomua with Chbyshv s poynomias. Th sut is psntd by th bu cuv. Th agmnt with th pvious cuv is quit good. Th diffnc again can b xpaind by two asons: in aity, th potntia is not piodic and th division is quit ough. 37

38 5 Gauss chain, N5,h m, a. mb.95 cm, L8 m, k m Ruk, Imuk, uk,, m , m Fig. 4.5 a. Imuk,. Imuk,, m , m Fig. 4.5 b Fig. 4.5: Potntia u k, fo th piodica wiing stuctu. a a and imaginay pats, b imaginay pat. 38

39 Ruk,, m Ra pat of th "potntia" fo Gauss Chain fo diffnt fquncis k k.37 k.88 k.44 k.59 k.743 k.89 k.4 k.9 k.34 k , m Fig. 4.6 a Imagin pat of th "potntia" fo Gauss chain potntia fo diffnt fquncis.. Imuk,, m k k.37 k.88 k.44 k.59 k.743 k.89 k.4 k.9 k.34 k , m Fig. 4.6 b Fig. 4.6: Spatia dpndnc of th a and imaginay pat of th patia potntia fo th quasi-piodica Gauss systm th cnta piod, n, 4 z 4 fo diffnt fquncis. 39

40 .. Cacuatd dpndnc Imuk, Quadatic appoximation Imuk,~ -.376*k -. Imuk,, m k, m - Fig. 4.7 a.3 Cacuatd dpndnc Imuk,.3 m Quadatic appoximation Imuk,.3 m~.4*k.5 Imuk,.3 m, m k, m - Fig. 4.7 b Fig. 4.7: Fquncy dpndncis xact and appoximation of th imaginay pat of th patia potntia fo th quasi-piodica Gaussian systm th cnta piod, n fo diffnt spatia points: a - ; b -. 3 m. 4

41 . Th invstigatd "Gaussian pit" wiing systm h m, h.5 m U V, Z c 37,7 Ω k ~U ZC y5 /U.8 x, m Z C U z, m h Z C Fig. 4.8: Gomty of th quasi - piodica wiing stuctu fo th CONCEPT simuation. 4

42 5 On "potntia" of th Gauss chain, N5,h m, b.95 cm, a. m, L8 m, k m - 4 uk,, m - 3 Ruk, Imuk, , m Fig. 4.9: On patia potntia cnta of th Gaussian wiing chain 4 z 4. Rfction and popagation cofficints Rk - fction cofficint k - tansmission cofficint k, m - Fig. 4.: Rfction and tansmission cofficints fo th on patia potntia cnta of th Gaussian wiing chain. 4

43 . CONCEPT Tans. matix-appox, ow-fquncy Tans. matix-appox, High-fquncy, fu intva Tans. matix-appox, Smi-anaytica..8 k k, m - Fig. 4.: Tansmission cofficint fo th considd piodica stuctu cacuatd by diffnt mthods. Now w consid ohmic osss and dictic coating on th popagation of cunt wavs though th quasi-piodica wiing stuctu. Fist, w consid th ohmic osss without adiation whn th potntia is dfind by th q Th numica cacuations hav shown that th a pat of th potntia 3.. pacticay coincids with th ossss cas s Fig. 4.6 fo k. Th imaginay pat of th potntia fo th diffnt conductivitis of th wi is shown in Fig. 4.. Th cosponding popagation cofficint is dispayd in Fig.4.3. On can obsv that th infunc of ohmic osss is sma in compaison with adiation osss. Scond, w invstigat th infunc of th dictic coating of th wi on th tansf cofficint. W consid a pfcty conducting wi with adius a cm coatd by a dictic ay with thicknss. 5 cm and dictic pmittivity of ε 3. Th cosponding potntia fo on sction cacuatd by 3.. in compaison with th uncoatd cas is psntd in Fig. 4.4, and th popagation cofficint is psntd in Fig Again, on can obsv that th infunc of th dictic coating on th popagation of th cunt wav is quit sma. 43

44 Imuk,, m σ5.76* 7 S/m Copp σ.3* 7 S/m Ion σ5.* 6 S/m σ.* 6 S/m σ5.* 5 S/m , m Fig. 4.: Imaginay pat of th potntia u k, fo on sction 4 z 4 of th piodica stuctu fo diffnt conductivitis of th wi k.4.. osss in σ5.76* 7 S/m Copp σ.3* 7 S/m Ion σ5.* 6 S/m σ.* 6 S/m σ5.* 5 S/m k, m - Fig. 4.3: Popagation cofficints though th piodica stuctu fo diffnt conductivitis of th wi. 44

45 5 4 3 non-isoatd wi coatd wi, 5 mm, ε3 u,, m , m Fig. 4.4: Th potntia u, fo on sction 4 z 4 of a pfcty conductiv coatd and uncoatd piodica wiing stuctu k.4. non-isoatd wi coatd wi, 5 mm, ε k, m - Fig. 4.5: Popagation cofficints though th pfct conductiv coatd and uncoatd piodica wiing stuctu. 45