Finite Element Analysis Of Left-handed Waveguides

Transcription

1 Univsiy of Cnal Floida lconic Thss and Dissaions Mass Thsis (Opn Accss) Fini lmn Analysis Of Lf-handd Wavguids 4 Balasubamaniam, Vllakkina Univsiy of Cnal Floida Find simila woks a: hp://sas.libay.ucf.du/d Univsiy of Cnal Floida Libais hp://libay.ucf.du Pa of h lcical and lconics Commons STARS Ciaion Vllakkina, Balasubamaniam,, "Fini lmn Analysis Of Lf-handd Wavguids" (4). lconic Thss and Dissaions. 56. hp://sas.libay.ucf.du/d/56 This Mass Thsis (Opn Accss) is bough o you fo f and opn accss by STARS. I has bn accpd fo inclusion in lconic Thss and Dissaions by an auhoid adminisao of STARS. Fo mo infomaion, plas conac l.doson@ucf.du.

2 FINIT LMNT ANALYSIS OF LFT-HANDD WAVGUIDS by SATISH VLLAKKINAR BALASUBRAMANIAM B.. Kumaaguu Collg of Tchnology A hsis submid in paial fulfillmn of h quimns fo h dg of Mas of Scinc in h Dpamn of lcical and Compu ngining in h Collg of ngining and Compu Scinc a h Univsiy of Cnal Floida Olando, Floida Fall Tm 4

3 ABSTRACT In his wok, wavguids wih simulanous ngaiv dilcic pmiiviy and magnic pmabiliy, ohwis known as lf-handd wavguids, a invsigad. An appoach of fomulaing and solving an ignvalu poblm wih fini lmn mhod suling in h dispsion laion of h wavguids is adopd in h analysis. Daild mhodology of ondimnsional scala and wo-dimnsional vco fini lmn fomulaion fo h analysis of goundd slab and abiay shapd wavguids is psnd. Basd on h analysis, fo wavguids wih convnional mdia, clln agmn of suls is obsvd bwn h fini lmn appoach and h adiional appoach. Th mhod is hn applid o analy lfhandd wavguids and anomalous dispsion of mods is found. Th disconinuiy sucu of a lf-handd wavguid sandwichd bwn wo convnional dilcic slab wavguids is analyd using mod maching chniqu and h suls a discussd basd on h inhn nau of h maials. Th scaing chaacisics of a paalll pla wavguid paially filld wih lf-handd and convnional mdia a also analyd using fini lmn mhod wih ignfuncion pansion chniqu. ii

4 ACKNOWLDGMNTS Fis and fomos, I wish o pss my dps gaiud o my adviso, D. Thomas X. Wu, fo his guidanc and valuabl advic fo h complion of his wok. His unabad nhusiasm, ncouagmn and suppo hav bn coninually fl and a galy appciad. I would lik o hank D. Pavn F. Wahid and D. Kalpahy B. Sundaam fo sving on my commi and fo shaing wih m hi pis and insigh. Spcial hanks o my pans fo hi lov, wisdom and painc. I would also lik o hank my finds and collagus who, on im o anoh, offd m hi much appciad hlp. Ths includ Hao Dong, Liping Zhng, and all oh finds in h Applid lcomagnics Goup a Univsiy of Cnal Floida. iii

5 TABL OF CONTNTS LIST OF FIGURS... vi CHAPTR : INTRODUCTION.... Ovviw.... Lf Handd Mdia: A Rviw....3 Compuaional Mhods Fini lmn Mhod Mod Maching Tchniqu Thsis Oganiaion... CHAPTR : ANALYSIS OF GROUNDD LH SLAB WAVGUID.... Ovviw.... Goundd LH Slab Wavguid....3 Fini lmn Analysis Rsuls and Discussion TM mods in a convnional goundd slab wavguid T mods in a convnional goundd slab wavguid TM mods in a goundd LH slab wavguid T mods in a goundd LH slab wavguid... 3 CHAPTR 3 : ANALYSIS OF DISCONTINUITIS IN LH SLAB WAVGUID Ovviw Scaing Chaacisics of Disconinuiis in LH Slab Wavguid Rsuls and Discussion iv

6 CHAPTR 4 : ANALYSIS OF PARALLL PLAT WAVGUID PARTIALLY FILLD WITH LH MDIA Ovviw Paalll Pla Wavguid Paially Filld wih LH Mdia Fini lmn Analysis Rsuls and Discussion CHAPTR 5 : ANALYSIS OF ARBITRARY SHAPD LH WAVGUID Ovviw Vco Fini lmn Fomulaion Fini lmn Analysis of a LH Wavguid wih Cicula Coss Scion Rsuls and Discussion Dispsion diagam of convnional wavguid wih cicula coss scion Dispsion diagam of LH wavguid wih cicula coss scion... 6 CHAPTR 6 : CONCLUSION APPNDIX A: RITZ VARIATIONAL PRINCIPL... 7 APPNDIX B: PROOF OF TH VARIATIONAL PRINCIPL APPNDIX C: GALRKIN MTHOD... 8 APPNDIX D: VCTOR IDNTITIS LIST OF RFRNCS v

7 LIST OF FIGURS Figu.: Wav popagaion chaacisics in (a) DPS mdium and (b) DNG mdium... 3 Figu.: Gomy of goundd LH slab wavguid... 3 Figu.: Goundd LH slab wih an lcical wall... 4 Figu.3: Dispsion diagam of TM mods in a convnional goundd slab wavguid wih ε =.55 and =... 9 Figu.4: Dispsion diagam of T mods in a convnional goundd slab wavguid wih ε =.55 and =... Figu.5: Dispsion diagam of TM mods in a goundd LH slab wavguid wih ε = -.55 and = -... Figu.6: Dispsion diagam of T mods in a goundd LH slab wavguid wih ε = -.55 and = ' Figu.7: Dispsion cuv of T mod in a goundd LH slab wavguid wih ε = -.55 and = Figu 3.: Goundd LH slab wavguid sandwichd bwn wo convnional goundd slab wavguids... 8 Figu 3.: Disconinuiy sucu du o symmy... 9 Figu 3.3: quivaln ansmission lin modl of h symmical disconinuiy sucu... 9 Figu 3.4: Pcnag of flcd pow of TM mod fo d =.3 λ wavguid hicknss Figu 3.5: Pcnag of ansmid pow of TM mod fo d =.3 λ wavguid hicknss Figu 3.6: Pcnag of flcd pow of TM mod fo d =.3938 λ wavguid hicknss Figu 3.7: Pcnag of ansmid pow of TM mod fo d =.3938 λ wavguid hicknss vi

8 Figu 3.8: Pcnag of flcd pow of TM mod fo d =.35 λ wavguid hicknss Figu 3.9: Pcnag of ansmid pow of TM mod fo d =.35 λ wavguid hicknss Figu 3.: Pcnag of flcd pow of TM mod fo d =.6 λ wavguid hicknss Figu 3.: Pcnag of ansmid pow of TM mod fo d =.6 λ wavguid hicknss... 4 Figu 3.: Pcnag of flcd pow of TM mod fo d =.6 λ wavguid hicknss... 4 Figu 3.3: Pcnag of ansmid pow of TM mod fo d =.6 λ wavguid hicknss... 4 Figu 4.: Schmaic of a paalll pla wavguid paially filld wih LH mdia Figu 4.: Paalll pla wavguid paially filld wih LH mdia and wih ficiious boundais Figu 4.3: Tansmid pow of a paalll pla wavguid paially filld wih convnional mdia... 5 Figu 4.4: Rflcd pow of a paalll pla wavguid paially filld wih convnional mdia... 5 Figu 4.5: Tansmid pow of a paalll pla wavguid paially filld wih LH mdia... 5 Figu 4.6: Rflcd pow of a paalll pla wavguid paially filld wih LH mdia... 5 Figu 5.: Abiay shapd LH wavguid wih lcical wall Figu 5.: Configuaion of (a) angnial dg lmns (b) nod lmns Figu 5.3: Schmaic of a LH wavguid wih cicula coss scion... 6 Figu 5.4: Cicula LH wavguid wih lcical wall... 6 Figu 5.5: Dispsion diagam of convnional wavguid wih cicula coss scion... 6 Figu 5.6: Dispsion diagam of LH wavguid wih cicula coss scion vii

9 Figu 5.7: Dispsion cuv of ' H mod in a LH wavguid wih cicula coss scion ' Figu 5.8: Dispsion cuv of T and T mod in a LH wavguid wih cicula coss scion ' Figu 5.9: Dispsion cuv of TM and TM mod in a LH wavguid wih cicula coss scion viii

10 CHAPTR : INTRODUCTION. Ovviw In cn yas, h sudy of lcomagnic popis of compl maials wih simulanous ngaiv al pmiiviy and pmabiliy has aacd a lo of anion in sach. Ths mdia, ypically fd as lf-handd (LH) mdia, possss insing faus ha may lad o unconvnional phnomna in guidanc, adiaion, and scaing of lcomagnic wavs. Many sach goups a now ploing vaious aspcs of his class of compl mdia and sval ponial fuu applicaions hav bn spculad. Rcn sach advancmns hav povd h fasibiliy of aliing hs aificial maials, which w considd hypohical. Th novl and insing faus of hs maials and hi possibl applicaions o fuu dsign of nw dvics a h pimay moivaion of his sach wok. In his hsis, a comphnsiv fini lmn analysis of lf-handd wavguids is povidd. I is also h fis sudy of abiay shapd LH wavguid. Insad of h adiional appoach of solving h wavguid chaacisic quaions, a diffn appoach of using fini lmn mhod is followd o g h dispsion chaacisics of h wavguids. This appoach aks advanag of h inhn abiliy of h fini lmn mhod o solv an ignvalu poblm fo all possibl ignvalus. In his appoach an ignvalu poblm is fomulad fo h sucu o b analyd and hn dpnding on h gomy of h sucu, a on-dimnsional o wo-dimnsional fini lmn pogam is dvlopd o solv h poblm. Basd on his mhodology, h goundd slab wavguid wih convnional mdia is analyd using on-dimnsional scala fini lmn fomulaion and h suls a compad wih hos

11 obaind fom h adiional mhod and an clln agmn is found. Thn h goundd LH slab wavguid is analyd and h anomalous dispsion chaacisics a obaind. A disconinuiy sucu, wih a LH slab wavguid sandwichd bwn wo dilcic slab wavguids, is invsigad using h mod maching chniqu and h physical insighs of h wav inacion bwn h wo mdia a psnd. Th scaing chaacisics of a paalll pla wavguid paially filld wih LH and convnional mdia a analyd using fini lmn mhod wih ignfuncion pansion chniqu. A obus, wo-dimnsional vco fini lmn fomulaion fo h analysis of abiay shapd wavguid sucus is psnd in dail. Th mhod is hn applid o analy convnional and LH wavguids wih a cicula coss scion. Fom h analysis, h is clln agmn of h suls fo h convnional wavguid and anomalous dispsion is nod in LH wavguids. Th anomalous dispsion chaacisics of LH wavguids a compad in associaion wih h convnional wavguids and h physical insighs a discussd basd on h inhn nau of h maials.. Lf Handd Mdia: A Rviw lcomagnic popis of maials can b chaacid by dilcic pmiiviy (ε) and magnic pmabiliy (). Popagaion popis of lcomagnic wavs in a maial a dmind by ε and ; which gula h laionship bwn h lcic fild and magnic fild. Wav popagaion in a hypohical maial wih simulanous ngaiv dilcic pmiiviy and magnic pmabiliy was fis nvisiond in h 96 s [] by h Russian physicis Vslago. H pdicd ha lcomagnic wav popagaion in hs mdia should giv is o sval pculia chaacisics. Accoding o Mawll s quaions, h dicion of ngy flow of a plan wav is givn by h dicion of h Poyning vco (S), which is h

12 coss poduc of lcic fild () and magnic fild (H). Fo plan wavs popagaing in isoopic gula mdia having simulanously posiiv ε and (doubl posiiv (DPS) mdium), h coss poduc of lcic fild () and magnic fild (H) givs boh h dicion of ngy flow (h Poyning vco) and h wav islf (ha is, h phas vlociy, o wav vco), and, H and wav vco k fom a igh-handd ipl of vcos. Bu Vslago pdicd ha in a mdium having simulanously ngaiv ε and (DNG mdium), whil H fo a plan wav sill givs h dicion of ngy flow, h phas vlociy (wav vco) shall b in h opposi dicion of ngy flow, and, H and wav vco k shall fom a lf-handd ipl of vcos. Figu. (a) and (b) shows h igh-handd wav popagaion bhavio in a DPS mdium and h lf-handd wav popagaion bhavio in a DNG mdium, spcivly. k S k S H H (a) (b) Figu.: Wav popagaion chaacisics in (a) DPS mdium and (b) DNG mdium Du o his lf-handd chaacisic, Vslago md such yp of maials as lfhandd mdium (LHM), and all gula maials w cospondingly md igh-handd 3

13 m dium (RHM). In addiion o his lf-handd chaacisic, LHM maials hav sval oh damaically diffn lcodynamic popis compad wih gula maials, smming fom a simulanous chang of h signs of ε and, including anomalous facion, vsal of boh h Doppl shif and h Chnkov adiaion, and vsal of adiaion pssu o adiaion nsion []. Alhough hs couninuiiv popis follow dicly fom Mawll s quaions, du o h absnc of naually occuing maials having simulanously ngaiv ε and, Vslago s pdicion was considd as a hoical concp and h has bn lil ffo o b undsand h lcomagnic bhavios of hs maials fo almos h dcads. Fo LH maials, sval oh nams and minologis hav also bn suggsd, such as mdia wih ngaiv faciv ind, backwad wav mdia (BW mdia), doubl ngaiv (DNG) mdia, and ngaiv ind mdia (NIM), o nam a fw. Th cn sugnc of ins in his mdium bgan whn Pndy suggsd is fis applicaion []; h poposd h possibiliy ha a pfc lns can b mad by using LH m dium which wih a ngaiv ind of facion migh ovcom known poblms wih common lnss. Ziolkowski invsigad h popagaion of lcomagnic wavs in LH mdia fom boh analyical and numical poins of viw. His analyical soluion fo a machd LH slab dmonsad ha h Pndy pfc lns ffc could b alid only in h psnc of a nondispsiv, losslss LH mdium [3]. Th lns ffc was shown no o is fo any alisic dispsiv, lossy LH mdium. Inspid by h wok of Pndy, Smih, Schul and Shlby consucd such a composi mdium by aanging aays of small mallic wis and spli ing sonaos [4, 5]. This discovy aousd ga ins in h unusual lcodynamic popis of LHM. Ziolkowski considd LH maials compisd of a subsa wih mbddd 4

14 capaciivly loadd sips and spli ing sonaos and dmonsad ha LH maial can b dsignd, fabicad, and sd wih micowav ngining ools [6]. ngha poposd h LH wavguid and analyd h dispsion diagam of LH slab wavguid, and found ha h poion of h guidd mod insid h slab has h Poyning vco ani-paalll o h dicion of phas flow of h mod and h poion of guidd mod ousid h slab has h Poyning vco paalll wih phas flow [7]. Alu and ngha invsigad h mod coupling bwn a convnional slab wavguid placd n o a LH slab wavguid and dmonsad h ani-dicional coupling nau hibid bwn h lays [8]. Thy also analyd vaious popis of h guidd mods in paalll pla wavguid filld wih pais of lays mad of any wo of h losslss psilon-ngaiv, mu-ngaiv, DPS, and DNG maials [9]. In h analysis, hy usd h chaacisic quaion of h wavguid o g h dispsion laion of diffn maial combinaions [9]. Basd on h analysis, hy suggsd som ponial applicaions in h dsign of novl dvics and componns, such as ula-hin wavguids, singl-mod hick fibs wih lss sicion and mo flibiliy on h fib hicknss, and vy hin caviy sonaos. Th analysis of guidd mods in a LH goundd slab and is possibl applicaion as novl subsas fo micosip annnas and aays is psnd in []. Ziolkowski and Kippl invsigad an lcically small dipol annna wih a shll of LH maial and found ha poply dsignd dipol-lh shll combinaion incass h al pow adiad by mo han an od of magniud ov h cosponding f spac cas []. Basd on h wavguid chaacisics quaion, Nfdov and Tyakov discussd h influnc of lay hicknsss and maial paams ov h wavguid popagaion chaacisics of vaious mods []. Fom h cn sach wok i looks vy pomising ha hs LH maials will hav a song 5

15 impac on h chnological wold. Much sach ploing h oic popis of hs maials is undway and many fuu applicaions hav bn spculad. Howv, h dispsion chaacisics of abiay shapd LH wavguids hav no bn analyd o da..3 Compuaional Mhods lcomagnic analysis in many ngining and scinific disciplin is basd on h lcomagnic pincipls govnd by h wll known s of quaions fomulad by Jams Clk Mawll in 873. lcomagnic fild poblms ais in many aas such as lcical machins, communicaion sysms and lconics, h compl gomy and high accuacy quimns of hs poblms mak i inviabl o sk numical soluion. Many highly sophisicad numical mhods hav bn dvlopd ov h yas and nw chniqus a always bing inoducd. A compuaional lcomagnic mhod may b viwd as a compuaional algoihm capabl of solving Mawll s quaions subjc o appopia bounday consains in a gnal configuaion and in h psnc of maials wih diffn popis. Cunly availabl compuaional mhods includ h Fini lmn Mhod (FM) [3, 4], Fini Diffnc Tim Domain mhod (FDTD), Mod Maching Tchniqu (MMT), Tansmission Lin Mhod (TLM) and Mhod of Momns (MoM), Mon Calo Mhod (MCM), Mhod of Lins (MOL) [4]. Ths mhods a usd o solv poblms ha a psnd by mans of diffnial quaions and oh mahmaical foms. Th innumabl sach publicaions and h numb of sofwa packags vify h impoanc of hs chniqus in solving vaiy of poblms. 6

16 .3. Fini lmn Mhod Th oigin of Fini lmn Mhod (FM) das back o 943 whn Couan publishd his wok manuscip of h addss h dlivd o h Amican Mahmaical Sociy [5]. In is pimiiv sag of dvlopmn, FM was usd as a sucual analysis ool o hlp aospac ngins in dsigning b aicaf sucus. La on, aidd by h apid incas of compu pow, h mhod has bn coninually dvlopd unil i bcam a vy sophisicad gnic ool fo accomplishing a wid aay of ngining asks. Is dvlopmn and succss is no paallld by any oh numical analysis chniqu. Th chniqu is basd on h pmis ha an appoima soluion o any compl ngining poblm can b achd by subdividing h poblm ino small mo managabl (fini) lmns. Using fini lmns, solving compl paial diffnial quaions ha dscib h bhavio of cain sucus can b ducd o a s of lina quaions ha can asily b solvd using h sandad chniqus of mai algba. Alhough h ali mahmaical amn of h FM was povidd in 943, h mhod was no applid o lcomagnic poblms unil 968. Sinc hn h mhod has gaind impoanc and h sysmaic gnaliy of h mhod maks i possibl o consuc gnal pupos compu pogams fo solving a wid ang of poblms in divs aas such as wavguid poblms, lcic machins, micosips and smiconduco dvics. In FM, accoding o h gomy of h poblm, h ni sucu domain is dividd ino sval subdomains, calld lmns. Thn h unknown funcions in ach lmn a pandd by h nodal valus (which a valus of h funcions in som paicula poins (nods o dgs) of h lmn), and h cosponding shap funcions (inpolaion funcions) [3]. Thus h oiginal poblm wih an infini numb of dgs of fdom is convd ino a 7

17 poblm wih fini numb of dgs of fdom. Thn using Rayligh-Ri o Galkin pocdu [3, 6], a sysm of algbaic quaions is fomd fom which a submai of ach lmn can b obaind. ach im whn a submai of a nw lmn is obaind i is assmbld wih h ising sysm mai. Finally, whn all h lmns a calld in on gs a sysm mai quaion fom which h nodal o dg valus a solvd. Sinc is fis applicaion o classical lcomagnic poblm of guidd popagaion [7] many impovmns hav nhancd h accuacy, gomical modling capabiliy and fficincy of FM. Also whn combind wih oh chniqus (lik Gn s funcion, ignfuncion pansion mhod, absobing bounday condiion) FM can, in gnal, ackl abiay gomis wih inhomognous, anisoopic, and/o lossy mdium. Consqunly, compu pogams dvlopd fo a paicula disciplin hav bn applid succssfully o solv poblms in a diffn fild wih lil o no modificaion. Wih h spasiy of h cofficin maics, FM hibis h ah plasing chaacisic of compuaional conomy in numical modling. Th succss of FM in lcomagnics can b lagly aibud o hi ga vsailiy and flibiliy, which allow h amn of gomically compl sucus wih inhomognous anisoopic o vn nonlina maials..3. Mod Maching Tchniqu Th concp of maching mods in lcomagnic analysis was fis poposd by Wl in h 967 [8]. Th usfulnss and accuacy of his mhod assud i a ga dal of anion fom sachs sinc ha im [9, ]. Du o h limid compu pow availabl a ha im i was no possibl o do mo han simpl numical compuaions. Th compuaional 8

18 mphasis was on ducing h numb of mods o h minimum so ha a numical soluion could b obaind. I was h aival of powful compus ha nabld h concp o b applid o h analysis of complicad sucus. Mod Maching Tchniqu (MMT) is a powful mhod fo h analysis and dsign of many lcomagnic componns and dvics. Th ssnc of h mhod is o divid h compl sucu o b analyd ino small scions and o mach h oal mod filds a ach juncion bwn scions. In analying a sucu using MMT, h fild in h sucu is pandd in a compl s of vco wav funcions, hs funcions a usually dnod as mods. Th mods which can popaga in ach scion, including in som cass vanscn mods, a s up and h ampliuds of h mods a machd acoss h bounday of h scion o h n scion wih appopia amn of h bounday condiions. Th ampliuds of mods a h oupu of a juncion can b dducd in ms of h ampliuds of h mod spcum a h inpu o h juncion. Th sngh of MMT sms fom h fac ha h ampliuds of h mods can b pssd as h componns of a scaing mai. ach juncion along h sucu has is own scaing mai. Th maics fo all juncions can b cascadd and suls in an ovall scaing mai fo h sucu. Th pocss of compuing h ovall scaing mai can b dcoupld o obain h scaing mai of paicula scion. Du o is numical fficincy and obusnss, mod maching analysis has bn widly mployd fo dsigning micowav componns. 9

19 .4 Thsis Oganiaion In his hsis, w po h modling and analysis of LH wavguids using FM and MMT. In od o mphasi h unusual and oic popis of LH mdia, w compa hm wih h convnional mdia as wll. Th chnical analyss a conaind in Chap hough Chap 5 and hi associad appndics. In chap, w illusa h mhodology o fam an ignvalu poblm fo h goundd LH slab wavguid and h on-dimnsional scala (nod basd) fini lmn fomulaion in dail. W po h dispsion chaacisics of h mods fo h convnional goundd slab in od o valida h accuacy of h fini lmn mhod. Thn h anomalous dispsion chaacisics of h mods in a goundd LH slab wavguid a pod. A daild discussion of h dispsion chaacisics of boh h wavguids is povidd and h unusual bhavio of LH wavguids is laboad. In chap 3, w povid h mod maching analysis of h disconinuiy sucu of a LH wavguid sandwichd bwn wo dilcic slab wavguids. Considing h symmical popis of h sucu, h scaing chaacisics of disconinuiis in h longiudinal dicion a invsigad using quivaln ansmission lin modl. Basd on his analysis, fou cass wih diffn wavguid highs a discussd and lvan physical insighs a psnd. In chap 4, h ignfuncion pansion chniqu in FM is plaind and h mhodology o solv unboundd fild poblms is illusad. Basd on his chniqu, h scaing chaacisics of a paalll pla wavguid paially filld wih LH and convnional mdia a analyd.

20 In chap 5, w povid h in-dph dails of h wo dimnsional vco fini lmn fomulaion fo h analysis of abiay shapd wavguid sucus. Th mhod is hn applid o analy convnional and LH wavguids wih a cicula coss scion. Fom h analysis, h anomalous dispsion chaacisics of LH wavguids a compad in associaion wih h convnional wavguids and h physical insighs a discussd basd on h inhn nau of h maials. A summay of conclusions of his sach wok is givn in chap 6.

21 CHAPTR : ANALYSIS OF GROUNDD LH SLAB WAVGUID. Ovviw Th main concn of his sach is o analy h dispsion chaacisics of LH wavguids, ohwis known as DNG wavguids. This chap povids an oulin on goundd LH slab wavguid and h on-dimnsional scala fini lmn appoach o obain is dispsion chaacisics. Basd on fini lmn analysis, h dispsion chaacisics fo convnional goundd slab wavguid and goundd LH slab wavguid a compad. Th anomalous dispsion in LH wavguid is discussd in dail.. Goundd LH Slab Wavguid Dilcic slabs and ods, wih o wihou any associad mal, a usd o conain h ngy associad wih a wav wihin a givn spac and guid i in paicula dicion. Typically hs a fd o as dilcic wavguid, and h fild mods ha hy can suppo a known as sufac wav mods []. Sufac wavs a psnd by a fild ha dcays ponnially away fom h dilcic sufac, wih mos of h fild conaind in o na h dilcic. A high fquncis h fild gnally bcoms mo ighly bound o h dilcic, making such wavguids pacical []. A goundd LH slab wavguid is a dilcic yp of wavguid ha has a gound plan covd wih a dilcic slab of high d, as shown in Figu., and ngaiv laiv pmiiviy ε and laiv pmabiliy. Th objciv of h wavguid o conain h ngy wihin h sucu and dic i owad a givn dicion is accomplishd by having h wav

22 bounc back and foh bwn is upp and low infacs a an incidnc angl ga han h ciical angl. Whn his is accomplishd, h facd filds ousid h dilcic fom vanscn (dcaying) wavs and all h al ngy is flcd and conaind wihin h wavguid. Th chaacisics of his wavguid can b analyd by aing h sucu as a bounday-valu poblm whos modal soluion is obaind by solving h wav quaion and nfocing h bounday condiions. Ai d ε < < Gound plan Figu.: Gomy of goundd LH slab wavguid Fo a goundd LH slab h is wo ss of disinc sufac mods. On s of mods has no magnic fild componn in h popagaion dicion; hs mods a fd as ansvs magnic (TM) mods and h oh s of mods has no lcic fild componn in h popagaion dicion; hs mods a fd o as ansvs lcic (T) mods. 3

23 .3 Fini lmn Analysis Fo a goundd LH slab, in addiion o h guidd wav mods, h adiaion and vanscn wavs compis a coninuous spcum. W plac an lcical wall a a high h abov h sucu o disci h coninuous spcum. Fo h goundd LH slab wavguid, as shown in Figu., h TM and T mods can b considd spaaly. lcical wall h Ai d ε < < Gound plan Figu.: Goundd LH slab wih an lcical wall Fom souc f Mawll s cul quaion, w obain = j ω H (.a) H = jωε ε (.b) wh ε and a h laiv pmiiviy and pmabiliy. 4

24 Fo h TM mods, w hav o y j H ε ωε = (.a) y o H j ω = (.b) o y j H ε ωε = (.c) Fom h abov quaions (.a-.c) w can g h Hlmhol wav quaion fo H y, = + y y H k k H ε ε (.3) wh k is h f spac wavnumb. Similaly fo h T mods, w hav o y H j ω = (.4a) y o j H H ε ωε = (.4b) o y H j ω = (.4c) Fom h abov quaions (.4a-.4c), h Hlmhol wav quaion fo y can b dducd as, = + y y k k ε (.5) quaions (.3) and (.5) hav h fom of h gnalid ignvalu poblm. Th dispsion chaacisics of h goundd LH slab wavguid can b obaind by solving hs ignvalu poblms. This class of poblms can b dal wih using h fini lmn mhod, and h sulan sysm of quaions has h fom of h gnalid ignvalu quaion. 5

25 [ ]{ φ} λ[ B]{ φ} = { } A (.6) wh [A] and [B] a known maics and λ and { φ } a unknowns. In h fini lmn mhod w will solv fo h ignvalu λ, which maks h sysm singula, o in oh wods, which maks h dminan of [A- λ B] vanish. As a sul, h will b a cosponding nonivial soluion fo { φ } which is calld ignvco. Th fis sp and mos impoan sp in h fini lmn analysis is h disciaion of h domain. In his sp h soluion domain Ω, ha is (, h), is subdividd ino numb of small domains usually fd o lmns. Du o on-dimnsional nau of h domain, h lmns a sho lin sgmns of lngh l ha a inconncd o fom h oiginal lin. Th poblm is fomulad in ms h unknown funcion{ φ }, ha is y o H y, a wo nods associad wih h lina lin lmn. Th scond sp is h slcion of an inpolaion funcion ha povids an appoimaion of h unknown soluion wihin ach lmn. Fo h lina lmn, a lina inpolaion funcion is slcd which a nono wihin h lmn and vanish ousid h lmn. Onc h inpolaion funcion is slcd, w can div an pssion fo h unknown soluion in an lmn, say lmn, in h following fom, φ = n j= N j φ j (.7) wh n is h numb of nods in h lmn, N j is h inpolaion funcion fo nod j. φ j is h valu of φ a nod j of h lmn, and Wih h pansion of φ givn in quaion (.7) w can fomula h sysm of quaions using Ri vaiaional mhod, in which an quivaln vaiaional poblm is 6

26 fomulad. A daild pocdu on h fomulaion of h vaiaional poblm is illusad in Appndi A. Th cosponding funcional fo h quaions (.3) and (.5) is givn by ( ) d k k F h = φ ε φ ε φ fo TM mod (.8a) ( ) d k k F h = φ ε φ φ fo T mod (.8b) Wihou loss of gnaliy h fini lmn mhod is plaind wih TM mod. Th funcional can b win as, (.9) ( ) ( ) = = M F F φ φ wh M dnos h oal numb of lmns and F is h subfuncional fo h h lmn givn by ( ) ( ) d k k F l = φ ε φ ε φ (.) wh l is h lngh of h h lmn. Inoducing pssion (.7) fo and diffniaing F φ wih spc o yilds φ i ( ) ( ) d N N k N N k N N F l j i j i j i j j i + = = ε ε φ φ i =, (.) In mai fom, his can b win as [ ]{ } K F φ φ = (.) wh T F F F = φ φ φ ; { } { } T φ φ φ = 7

27 Sinc is h unknown ha has o b solvd, w spli K k ino wo pas, [ ]{ } [ ]{ } B k A F φ φ φ = (.3) Th lmns of h maics [A ] and [B ] a givn by ( ) d N N k N N A l j i j i ij + = ε (.4a) ( ) d N N B l j i ij = ε (.4b) Th lmnal quaions (.3) a assmbld fo all M lmns and h saionay quimn on F is imposd o find h sysm of quaions [ ]{ } [ ]{ } ( {} = = = = = M M B k A F F φ φ φ φ ) (.5) Th abov sysm of quaions can b win compacly as, [ ]{ } [ ]{ } φ φ B k A = (.6) his is cognid as h gnalid ignvalu poblm dfind in (.6). Solving h sysm of quaions wih appopia bounday condiions yilds h ignvalus o h longiudinal popagaion consans k. Th ffciv laiv dilcic consan ε of h LH slab is obaind fom h laion = k k ε (.7) Simila pocdu is followd fo T mods and h dispsion chaacisics of goundd LH slab wavguid a obaind. 8

28 .4 Rsuls and Discussion Basd on h fini lmn fomulaion dscibd abov h dispsion diagams of TM mods and T mods in a convnional goundd slab wavguid wih ε =.55 and = a obaind..4. TM mods in a convnional goundd slab wavguid Figu.3: Dispsion diagam of TM mods in a convnional goundd slab wavguid wih ε =.55 and = 9

29 Fom h dispsion diagam, shown in Figu.3, i can b sn ha fo any nono hicknss slab, wih a pmiiviy ga han uniy, h is a las on popagaing TM mod. This TM mod is h dominan mod of h dilcic slab wavguid and has a o cuoff fquncy. I can b sn ha h n TM mod, h TM mod, will no popaga unil h high of h slab bcoms ga.4λ. Also, all h dispsion cuvs incas monoonically..4. T mods in a convnional goundd slab wavguid Figu.4: Dispsion diagam of T mods in a convnional goundd slab wavguid wih ε =.55 and =

30 Fo T mods, fom h dispsion diagam i can b sn ha h fis mod dos no sa o popaga unil h high of h slab bcoms ga.λ. This T mod is h dominan mod of dilcic slab wavguid. All h T mods in h convnional dilcic slab hav cuoff fquncis. Fo h T mods h cuvs of ffciv dilcic consan ε vsus nomalid hicknss d/λ incas monoonically..4.3 TM mods in a goundd LH slab wavguid Th laiv pmiiviy ε and laiv pmabiliy a assumd ngaiv valus, ε = -.55 and = -, and h fini lmn analysis is caid fo goundd LH slab. Th dispsion chaacisics of TM mods in a goundd LH slab a shown in Figu.5. In od o undsand h physics of LH mdia, w compa h dispsion diagam of h convnional goundd slab wavguid and h goundd LH slab wavguid. In h convnional goundd slab wavguid, h TM mod has no cuoff fquncy and h dispsion cuv incas monoonically. On conas, fo h goundd LH slab wavguid all h mods hav cuoff fquncis. Moov, in h dispsion diagam h cuv no long incass monoonically, bu is bn in a spcial gion (shadd in Figu..5). Whn h high (d/λ ) of h slab is incasd fom, h is no popagaing mod in h goundd LH slab wavguid. Whn h valu of (d/λ ) achs.3938 (poin A in Figu.5), h fis TM mod appas. Fo planaoy pupos, his poin wh h fis mod appas is dfind as h ciical poin A. A h ciical poin, h poion of h guidd mod insid h goundd LH slab wavguid shows h Poyning vco o b ani-paalll o h dicion of h mod s phas flow and h

31 poion of his mod ousid h slab shows h Poyning vco o b paalll wih h phas flow. Bu h n pow is fo his cas. A 3 TM I A TM Spcial gion II A O III TM ' TM A ' TM ' TM Figu.5: Dispsion diagam of TM mods in a goundd LH slab wavguid wih ε = -.55 and = - Wih h incas of (d/λ ) byond h ciical poin, h is wo TM mods in h spcial gion. In his gion, h dispsion cuv gs bifucad in wo diffn dicions, ha is, cuv I is dcomposd ino cuv A A and cuv A A. Th ason fo calling i a spcial gion is as follows. Fo h cuv A A, h poion of h guidd mod insid h goundd LH

32 slab wavguid shows h Poyning vco o b ani-paalll o h dicion of h mod s phas flow and h poion of guidd mod ousid h slab shows h Poyning vco o b paalll wih h phas flow. Howv, in his cas h oal pow flow of h pa of h mod insid h slab is ga han ha of h oal pow flow of h pa of h mod ousid. In h oh wods, h n oal pow flow of h guidd mod is anipaalll wih h dicion of h phas flow. On h oh hand, fo h cuv A A, h poion of h guidd mod insid h goundd LH slab wavguid shows h Poyning vco o b anipaalll o h dicion of h mod s phas flow and h poion of guidd mod ousid h slab shows h Poyning vco o b paalll wih h phas flow. Bu h h oal pow flow of h pa of h mod insid h slab is small han ha of h oal pow flow of h pa of h mod ousid. In his cas, h n oal pow flow of h guidd mod is paalll wih h dicion of h phas flow, which is in conay o ha of cuv A A [3]. Whn h slab high (d/λ ) achs.49, h cuv A A in h spcial gion vanishs and h dispsion cuv I is ou of h spcial gion. Cuv A A 3 has h sam chaacisics as ha of cuv A A. In cuv I, h A A 3 poion ha has n ngaiv pow flow is dfind as TM mod and h AA poion ha has n posiiv pow flow is dfind as ' TM mod. A simila analysis can b don fo oh mods as wll..4.4 T mods in a goundd LH slab wavguid Th dispsion chaacisics of T mods in a goundd LH slab a shown in Figu.6. In a convnional goundd slab wavguid all h T mods hav cuoff fquncis and 3

33 h cuvs incas monoonically. On conas, h fis mod, T mod, in a goundd LH slab wavguid has no cuoff fquncy. Moov, in h dispsion diagam h cuvs no long incas monoonically, bu a bn in a spcial gion (shadd in Figu.6). A I Spcial gion T ' T II T 3 III ' T A A ' T 3 Figu.6: Dispsion diagam of T mods in a goundd LH slab wavguid wih ε = -.55 and = - Fom Figu.7, i can b sn ha whn h high (d/λ ) of h slab is incasd fom h T mod sas o appa. Also h dispsion cuv dcass monoonically. Fo his mod, h poion of h guidd mod insid h goundd LH slab wavguid shows h Poyning 4

34 vco o b anipaalll o h dicion of h mod s phas flow and h poion of guidd mod ousid h slab shows h Poyning vco o b paalll wih h phas flow. Th oal pow flow of h pa of h mod insid h slab is small han ha of h oal pow flow of h pa of h mod ousid. Hnc in his cas, h n oal pow flow of h guidd mod is paalll wih h dicion of h phas flow. Sinc his mod has n posiiv pow flow h mod is dfind as ' T mod, in associaion wih ou TM mod analysis. Fo his T mod, h is no ciical poin and also h is no n ngaiv pow flow gion in h dispsion cuv. Th T mod achs cuoff whn h valu of (d/λ) achs.989. ' ' T ' T Figu.7: Dispsion cuv of ' T mod in a goundd LH slab wavguid wih ε = -.55 and = - 5

35 In h dispsion diagam shown in Figu.6, i can b nod ha af h cuoff of mod h is no mod popagaion ill (d/λ ) achs Whn h (d/λ ) achs h n high od mod, T mod, sas o appa. A his ciical poin A, h poion of h guidd mod insid h goundd LH slab wavguid shows h Poyning vco o b anipaalll o h dicion of h mod s phas flow and h poion of his mod ousid h slab shows h Poyning vco o b paalll wih h phas flow. Bu h n pow is fo his cas. Af h ciical poin, h cuv II bifucas in wo diffn dicions in h spcial gion. Th n oal pow flow of h guidd mod in AA poion of h cuv is anipaalll wih h dicion of h phas flow, ha is, h n pow flow is ngaiv. Similaly h n oal pow flow of h guidd mod in A A poion of h cuv is paalll wih h dicion of h phas flow, ha is, h n pow flow is posiiv. Whn h slab high (d/λ ) achs.654, h cuv A A in h spcial gion vanishs and h dispsion cuv II is ou of h spcial gion. Th A A poion of cuv II is dfind as A simila bhavio is nod in T 3 mod as wll. ' T ' T mod and A A poion is dfind as T mod. 6

36 CHAPTR 3 : ANALYSIS OF DISCONTINUITIS IN LH SLAB WAVGUID 3. Ovviw This chap povids h fini lmn and mod maching analysis of h disconinuiy sucu of a LH wavguid sandwichd bw n wo dilcic slab wavguids. Th symmical popis of h sucu a considd in h analysis and h scaing chaacisics of disconinuiis in h longiudinal dicion a invsigad using quivaln ansmission lin modl. Basd on h analysis, fou cass wih diffn wavguid highs a discussd and lvan physical insighs a psnd. 3. Scaing Chaacisics of Disconin uiis in LH Slab Wavguid Disconinuiis in dilcic wavguids, which ais du o diffncs in dimnsions and/o maial popis of h mdium, hav bcom indispnsabl pa of many micowav componn dsign and ffos hav bn mad in h undsanding h flcion and ansmission phnomna a a disconinuiy infac. Th disconinuiy sucu of a goundd LH slab wavguid is sandwichd bwn wo convnional goundd slab wavguids is shown in Figu 3.. Th ffc of disconinuiy in h sucu shown in Figu 3. is an mly complicad on and his compliy aiss du o h lf-handd chaacisics of h mdia. 7

37 L Ai d ε > > ε < < ε > > Figu 3.: Goundd LH slab wavguid sandwichd bwn wo convnional goundd slab wavguids Th coninuous spcum of adiaion and vanscn wavs, in addiion o guidd wav mods, is discid by placing an lcical wall a a high h abov h sucu. Fo mod maching analysis of h disconinuiy sucu, w nd o find h ignvalus and ignfuncions of h guidd mods and h discid adiaion and vanscn mods in boh LH and dilcic wavguid. Du o h symmical nau of h disconinuiy sucu, as shown in Figu 3., h scaing o h guidd mods can b analyd in ms of h symmical and anisymmical ciaions fo which w hav h quivaln ansmission lin opn cicui and sho cicui spcivly, as indicad in Figu 3.3. Wihou loss gnaliy, h disconinuiy sucu is analyd wih TM mod. 8

38 lcical wall Ai L/ h d ε > > l ε < < Figu 3.: Disconinuiy sucu du o symmy L/ TM TM BANK TM TM ' O.C o S.C + Γ ( ) Γ ( ) l l Figu 3.3: quivaln ansmission lin modl of h symmical disconinuiy sucu 9

39 Mawll s cul quaion can b pssd as = j ω H (3.a) H = jωε ε (3.b) Fo h TM mods, w hav H y = jωε ε o (3.a) = jω H o y (3.b) H y = jωε ε o (3.c) Fom h abov quaions, w can also obain wh h fild componns a pssd as ( ) U ( ) = (3.3a) ( ) J ( ) = (3.3b) ( ) J ( ) H y = hy (3.3c) ( ) ε = ε Acos ( k ), A cos d [ k ( h ) ], d h (3.4a) ( ) h y k A sin jωε = k Asin jωε A cos ( ) ( k ), = Acos ( k ), [ k ( h ) ], [ k ( h ) ], d d h d d h (3.4b) (3.4c) 3

40 In h quaions (3.3a-3.3c), U () and J () saisfy h ansmission lin quaions du d ( ) ( ) dj d wh h chaacisic impdanc Z c is givn by ( ) = jk Z J (3.5a) c ( ) = jk Y U (3.5b) c k Z c = = (3.6) Y ωε c Basd on h fini lmn appoach discussd in chap, h ignvalus o h popagaion consans and h cosponding ignfuncions o h fild componns of h convnional slab and LH slab wavguids a dmind. Bfo h mod maching amn in h longiudinal dicion, h ignfuncions of ach mod should b nomalid by S, wh h S = h d (3.7) y Th gnal fild soluion in ach unifom gion may b pssd in ms of h supposiion of a compl s of mod funcions. Hnc h angnial fild componns in h < l gion a pssd as = (, ) n ( ) U n( ) (3.8a) = n y, = hyn J n n= ( ) ( ) ( ) H (3.8b) and h angnial fild componns in h > l gion a givn by (, ) ( ) U ( ) = n n (3.9a) n= 3

41 (, ) h ( ) J ( ) = H y yn n (3.9b) n= A h sp disconinuiy a = l, h angnial fild componns mus b coninuous saisfying h condiions n= n ( ) U ( ) = ( ) U ( ) n (3.a) n= n ( ) J ( ) = h ( ) J ( ) yn n n= n= Th fild componns saisfy h following ohogonaliy laion, yn n h (3.b) h n nhymd = (3.) nm his yilds U = Q U (3.a) wh Q and P a h coupling maics givn by ( l ) = mn l J = P J (3.b) h l Q h d (3.3a) ( l ) = mn n h m ym P h d (3.3b) Making us of h ohogonaliy laion, quaion (3.), w can also obain T P l U = U (3.4a) T Q l J = J (3.4b) yn 3

42 Fom quaions (3.) and (3.4), w can div P T l Ql T = Ql Pl = (3.5) wh is h uni mai. Fo h ansmission lin modl, shown in Figu 3.3, h inpu impdanc mai a = l plan, looking o h igh of h sucu, saisfis T ( ) Q Z( ) Z = + Q (3.6) l l l l and h flcion cofficin mai a h = l plan, looking o h igh, is pssd as Γ ( ) [ Z( ) + Z ] [ Z( ) Z ] l = (3.7) l c Wih h biscions in h longiudinal dicion du o h symmical nau of h disconinuiy sucu, h a wo diffn combinaions of bounday condiion. Fo an incidn guidd mod fom h inpu wavguid, convnional slab wavguid, wo spaa subsucus a analyd wih hi spciv bounday c ondiions. In ach cas, ngy is flcd and h flcion cofficin maics a dnod by R o and R s fo h quivaln ansmission lin opn cicui and sho cicui, spcivly. Wih h flcion maics Ro and R of h subsucu, h flcion cofficin mai R and ansmission cofficin mai T s of h ni sucu a dmind by [4, 5] ( R + R ) l c o s R = (3.8a) ( R R ) o s T = (3.8b) 33

43 3.3 Rsuls and Discussion Basd on h mod maching appoach h disconinuiy sucu of a LH wavguid sandwichd bwn wo dilcic slab wavguids is analyd fo diffn wavguid highs. Th TM mod wih incidn pow of is assumd o b incidn fom h slab wavguid and h pcnag pow of h flcd and ansmid mod is calculad fo fou diffn cass. Cas : d =.3 λ In his cas, h is only h dominan TM mod in h convnional goundd slab wavguid and all h mods a cuoff in h LH slab wavguid. Fom Figu 3.4, i is nod ha whn h nomalid lngh (L/λ ) of h LH slab wavguid is lss han wo, h is coupling bwn h wo convnional slab wavguids. Figu 3.4: Pcnag of flcd pow of TM mod fo d =.3 λ wavguid hicknss 34

44 Bu whn h nomalid (L/λ ) incass abov wo, h coupling ffc vanishs and h pcnag pow of h flcd and ansmid TM mod bcom consan. Bcaus all h mods a cuoff in h goundd LH slab wavguid, fom Figu 3.5, h pcnag pow of h ansmid TM mod is. This is h cas whn h convnional dilcic slab wavguid is conncd wih an infinily long LH slab wavguid. Figu 3.5: Pcnag of ansmid pow of TM mod fo d =.3 λ wavguid hicknss Cas : d =.3938 λ In his cas, h high of h wavguid cosponds o h ciical poin on h dispsion cuv of h LH slab wavguid, shown in Figu.5. Fo his high only on TM 35

45 mod wih o n oal pow appas in h LH slab wavguid and only h dominan TM mod iss in h convnional goundd slab wavguid. Figu 3.6: Pcnag of flcd pow of TM mod fo d =.3938 λ wavguid hicknss Fom Figu 3.5 and Figu 3.6, i can b infd ha 5% pow of h TM mod is flcd and h is small ansmission. Also, fo spcific (L/λ ) valus of, 44, 66 and 88, song sonanc occus. Ths nomalid lnghs cospond o h sonan fquncis of h LH sonao which has h sam sucu as shown in Figu 3.. W can find ha a hs sonan poins, h TM mod has no flcion and mos of h pow is ansmid in h longiudinal dicion. 36

46 Figu 3.7: Pcnag of ansmid pow of TM mod fo d =.3938 λ wavguid hicknss Cas 3: d =.35 λ Fo his cas, h is only TM mod in h convnional goundd slab wavguid. Fo h goundd LH slab wavguid, his high is in h spcial gion as shown in Figu.5. Hnc h a wo TM mods, h TM mod wih n ngaiv oal pow flow and ' TM mod wih n posiiv oal pow flow. Ths suls a shown in Figu 3.8 and Figu 3.9. Fom h cuvs, w find ha h a oscillaions in h flcd and ansmid pow which is du o h coupling of TM and ' TM mods. Compad wih cas, h ansmission of TM mod aks h majoiy and h flcion is much small. 37

47 Figu 3.8: Pcnag of flcd pow of TM mod fo d =.35 λ wavguid hicknss Figu 3.9: Pcnag of ansmid pow of TM mod fo d =.35 λ wavguid hicknss 38

48 Cas 4: d =.6 λ In his cas, h convnional goundd slab wavguid has TM and TM mods. In h LH slab wavguid, his poin is ou of h spcial gion in h dispsion cuv, only h TM mod iss. Th plos of h flcd TM and TM mod and ansmid TM and TM mod a shown blow. Figu 3.: Pcnag of flcd pow of TM mod fo d =.6 λ wavguid hicknss 39

49 Figu 3.: Pcnag of ansmid pow of TM mod fo d =.6 λ wavguid hicknss Figu 3.: Pcnag of flcd pow of TM mod fo d =.6 λ wavguid hicknss 4

50 Figu 3.3: Pcnag of ansmid pow of TM mod fo d =.6 λ wavguid hicknss Fom h abov figus, w can find ha mos of h pow of h TM mod is ansmid. Th flcion of h TM and TM mods and h ansmission of TM mod a all small. Th ason fo his o occu is as follows. In his cas, h ffciv dilcic consans of h TM and TM mods in h convnional slab wavguid a.43 and.3676 spcivly. F o h LH slab wavguid h ffciv dilcic consan of h TM mod is.348. Th TM mod in h convnional slab wavguid is closly machd wih h TM mod in h LH slab wavguid. Hnc wih h TM mod incidn, h ansmid TM mod is h majoiy componn and h flcd TM and TM mods and h ansmid TM mod a all small. 4

51 CHAPTR 4 : ANALYSIS OF PARALLL PLAT WAVGUID PARTIALLY FILLD WITH LH MDIA 4. Ovviw In his chap, h fini lmn mhod ogh wih ignfuncion pansion chniqu fo h analysis of a paalll pla wavguid paially filld wih LH mdia is illusad. I also povids h implmnaion of h mhod o analy a paalll pla wavguid paially filld wih convnional and LH mdia. Th scaing chaacisics of h paalll pla wavguid wih paial filling a obsvd and h insighs a discussd. 4. Paalll Pla Wavguid Paially Filld wih LH Mdia A paalll pla wavguid is h simpls yp of guid consising of wo paalll pfcly conducing plas ha ap popagaing ngy bwn hm. Th lcomagnic wavs insid bounc back and foh bwn h plas as h wav popaga down h wavguid so as o saisfy h pla bounday condiions. A paalll pla wavguid can suppo T, TM and TM mods. Figu 4. shows a paalll pla wavguid paially filld wih LH mdia. In h gomy of h paalll pla wavguid h mal pla widh is assumd o b much ga han h spaaion, a, so ha finging filds and any vaiaion can b ignod. Ai is assumd o fill h gion bwn h plas and a LH mdia of high d and widh w is placd in bwn h plas. 4

52 Mal pla Ai a w ε < < d Figu 4.: Schmaic of a paalll pla wavguid paially filld wih LH mdia 4.3 Fini lmn Analysis Th ignfuncion pansion chniqu wih h fini lmn mhod is widly usd in h amn of unboundd fild poblms [3]. In his mhod h unboundd gion is dividd ino an inio and an io gion, and h fini lmn mhod is mployd o fomula h fild in h inio gion. Th io filds a psnd by an pansion of ignfuncions, which maks his mhod diffn fom oh mhods. Th ignfuncions a a s of homognous soluions of a diffnial quaion saisfying cain bounday condiions. Thi pansions can b usd o psn a soluion of h cosponding inhomognous diffnial quaion wih any souc funcion, subjc o sam bounday condiions. This chniqu allows mulipl-mod popagaion and, mo impoan, allows h ficiious boundais o b placd as clos o h gion of ins as possibl. Ths a achivd by 43

53 including high-od mods in h psnaion of h filds a h ficiious boundais. This suls in a laivly small fini lmn disciaion and hby dcass h compuaion im. In h analysis of paalll pla wavguid paially filld wih LH mdia w disci h unboundd gion by nclosing h sucu wih wo ficiious boundais a = and = spcivly, shown in Figu 4.. Th filds in h inio gion (II) a fomulad using FM and h filds in h io gions (I and III) a pssd by pansion of ignfuncions. T I II III w Ai a ε < < d Figu 4.: Paalll pla wavguid paially filld wih LH mdia and wih ficiious boundais Fo h sucu shown in Figu 4., T mod is assumd o b incidn. Th fild a h ficiious plan a = can b win as h supposiion of h incidn fild and h fild flcd fom h LH mdia (disconinuiy). Sinc his ficiious plan is placd clos o h LH mdia (disconinuiy), h flcd fild canno b psnd by h dominan mod only; 44

54 ah, i is a supposiion of h dominan and many high-od mods cid by h disconinuiy. Thfo, w hav inc y (, ) = y (, ) + m= jkm a ( ) (4.) in which am a h pansion cofficins and ym () and k m a givn by, ym m ym mπ ( ) = sin( ) (4.) a a k m k = j mπ ( ) a mπ ( ) a mπ, if ( ) a m k, if ( k π ) a > k (4.3) wh a dnos h spaaion bwn h wo plas. Sinc ach mod givn by ( ) ym jkm is a homognous soluion of h Hlmhol quaion subjc o h bounday condiions applicabl o y, hi supposiion (4.) saisfis h Hlmhol quaion and h quid bounday condiion as wll. Fuh, sinc hs mods fom a compl s of ignfuncions, hy can psn any y disibuion wihin h wavguid. Using h ohogonaliy laion w obain h pssion fo h pansion cofficins a m in ms of h fild y as, a jk am m inc = y (, ) y (, )] ym( ) [ d (4.4) wh dnos h posiion of h ficiious plan. Insing his ino (4.), w obain a inc jkm ( ) inc (, ) = y (, ) + ym( ) [ y ( ', ) y ( ', )] ym( ') ' m= y d (4.5) Taking h paial diva iv of his quaion wih spc o yilds h bounday condiion a = which can b win in h gnalid fom as, 45

55 y (, ) + P[ y (, )] = U n inc (4.6) wh nˆ = ˆ, P y (, )] is h bounday opao givn by [ a )] = jk m ym ( ) y ( ', ) ym ( ') d' m= P [ (, (4.7) y and U inc is givn by inc (, ) a y jkmym( ) n m= inc inc U = ( ', ) ( ') d' (4.8) Similaly, h fild a = ficiious plan can b pssd as, y ym = y (, ) bm = m ym ( ) jk m (4.9) wh ym () and k m a also givn by (4.) and (4.3) spcivly. Fom (4.9), w can find a jkm bm = y (, ) ym( ) d (4.) and fuh h bounday condiion a = is givn by, y (, ) + P[ y (, )] = (4.) wh P[ (, )] is h bounday opao givn by quaion (4.7). y Wih bounday condiions (4.6) and (4.), h bounday-valu poblm fo h fild insid h fild insid h ficiious boundais is uniquly dfind. W can fomula h sysm of quaions using Ri vaiaional mhod, in which an quivaln vaiaional poblm is fomulad. A daild pocdu on h fomulaion of h vaiaional poblm is illusad in Appndi A. 46

56 Th funcional fo his poblm is pssd as, wh, F ) = F ( ) F ( ) F ( ) (4.) ( y y y 3 y y y F ( y ) = [( ) + ( ) ] k ε y dω (4.3a) Ω u a F ( y ) = yp[ y ] d = (4.3b) F a inc 3( y ) = ( yp[ y ] U y ) d = (4.3c) A daild poof fo h funcional is givn in Appndi B. Subsiuing h pssions fo P ] [ y inc and U ino h abov, w obain a funcional ha is amnabl o a fini lmn soluion. Wih h suiabl slcion of inpolaion funcion, w can div an pssion fo h unknown soluion in an lmn, say lmn, in h following fom, n φ = N jφ j (4.4) j= wh n is h numb of nods in h lmn, φ j is h valu of φ a nod j of h lmn, and N j is h inpolaion funcion fo nod j. Inoducing pssion (4.4) fo φ ino quaion (4.3a) and diffniaing h subfuncional F wih spc o φ i yilds F φ ) = [ A φ ( ]{ φ } k [ D i ]{ φ } (4.5) wh F F F F = φi φ φ ; { } { φ } T φ φ3 φ3 T φ = and h lmn maics [A ] and [D ] a givn by 47

57 A ij u = Ω Ni ( N j N + i N j ) dω (4.6) D ij = ε Ω ( N N i j ) dω (4.7) Following a simila appoach fo h oh wo subfuncionals w hav, F F ( φ ) = [ K φ ]{ φ } s i ( φ ) 3 = [ T ]{ } { s φ + bi φi } (4.8) (4.9) wh h lmn maics [ K ], [ T ] and h lmn vco {b } a givn by s s K s mπ sin( ) d + a s = jk m a W m= s s mπ sin( ) d a s mπ sin( ) sin( ) a d mπ W d + W a W (4.) T s K s = = (4.) b jk π = jk Wi sin( d a (4.) s a i ) Sinc h subfuncional F and F 3 psn h bounday condiion a h wo ficiious plans, W is assumd o b on-dimnsional lina inpolaion funcion. Th lmnal quaions a assmbld fo all M lmns and h saionaiy qu imn on F is imposd o find h sysm of quaions, Γ F = φ M F = φ = = M ( [ A ]{ φ } k [ D ]{ φ } [ K s ]{ φ } [ Ts ]{ φ } { b} ) = {} (4.3) 48

58 Th abov sysm of quaions can b win compacly as, [ A]{ φ} k [ D]{ φ} [ K]{ φ} [ T ]{ φ} { b} = (4.4) wh φ psns h unknown fild y. 4.4 Rsuls and Discussion Using h fini lmn fomulaion wih ignfuncion pansion chniqu, h scaing chaacisics of a paalll pla wavguid paially fill d wih convnional mdia and LH mdia a invsigad. Th suls a also compad wih mod maching chniqu as wll. Fis, h paalll pla wavguid is paially filld wih a convnional mdia wih pmiiviy ε =.55 and pmabiliy =. Th widh and high of h convnional mdia a assumd o b w = mm and d = 5.75mm spcivly. Fo a paalll pla wavguid wih a =.86mm, h cuoff fquncis of T and T mods a 6.56 GH and 3. GH spcivly. Th T mod wih pow of is assumd o b incidn fom h lf and h pcnag of flcd and ansmid pow is calculad. Fom Figu 4.3, i can b nod ha fom 8.5 GH o.5 GH h is compl pow ansmission. Also, h a som oscillaions bwn.5 GH and 3 GH which is du o h n high od mod in h paalll pla wavguid. Th pcnag of flcd pow is shown in Figu 4.4. Fom h scaing plos i can b sn ha h is an clln agmn of suls bwn h ignfuncion pansion chniqu and mod maching chniqu. 49

59 Figu 4.3: Tansmid pow of a paalll pla wavguid paially filld wih convnional mdia Figu 4.4: Rflcd pow of a paalll pla wavguid paially filld wih convnional mdia 5

60 Th sca ing plos of h paalll pla wavguid paially filld wih LH mdia wih pmiiviy ε = -.55 and pmabiliy = - a shown in Figu 4.5 and Figu 4.6. Th widh and high of h LH mdia is kp h sam as h convnional mdia cas and h pcnag of flcd and ansmid pow a calculad. Fom Figu 4.5, i can b sn ha h is no fquncy wih compl pow ansmission bu h avag pow ansmission ov h ni fquncy band is impovd. I can also b nod ha h high fquncy oscillaions sn in convnional mdia cas a no psn in LH mdia cas. Figu 4.5: Tansmid pow of a paalll pla wavguid paially filld wih LH mdia 5

61 Figu 4.6: Rflcd pow of a paalll pla wavguid paially filld wih LH mdia Fom Figu 4.5 and Figu 4.6, i can b sn ha h is an clln agmn of suls bwn h ignfuncion pansion chniqu and mod maching chniqu. Th paalll pla wavguid paially filld wih LH mdia has a wid ansmission band wihou oscillaions. 5

62 CHAPTR 5 : ANALYSIS OF ARBITRARY SHAPD LH WAVGUID 5. Ovviw This chap povids h wo-dimnsional vco fini lmn fomulaion fo h analysis of abiay shapd LH wavguid sucus. I also povids h implmnaion of h mhod o analy convnional and LH wavguids wih cicula coss scion. Th anomalous dispsion chaacisics of lf-handd wavguids a compad in associaion wih h convnional wavguids and h physical insighs a discussd. 5. Vco Fini lmn Fomulaion Th vco fini lmn mhod is widly usd o compu h mod spcum of an lcomagnic wavguid wih abiay coss scion [6]. Th vco fini lmn mhod is mos lgan and simpl appoach o limina h disadvanags of h scala fini lmn appoach. Th choic of dg basd lmns maks his mhod immun agains h undsid spuious mods o non-physical soluions and asy implmnaion of bounday condiions a maial infacs. In h analysis of any abiay shapd sucu, w disci h coninuous spcum by nclosing h sucu wih an lcical wall as shown in Figu