FIRST UNIVERSITY OF NAPLES FEDERICO II PHD SCHOOL IN: INNOVATIVE TECHNOLOGIES FOR MATERIALS, SENSORS AND IMAGING. XXII CYCLE ( ) THESIS

Transcription

1 FRST UNERSTY OF NAPES FEDERCO PHD SCHOO N: NNOATE TECHNOOGES FOR MATERAS, SENSORS AND MAGNG. XX CYCE (6-9 THESS NUMERCA-ANAYTCA METHODS FOR PARTCE ACCEERATORS TUTOR Pof. TTORO G. ACCARO CANDDATE MARCO PANNEO

2

3 Summay Summay noduion Chae : Main Paamee and Adoed Mehod. 3. Main Paamee The Wie Mehod...3 The Modal Exanion in a Cylindial Caviy Chae : The Pillox Caviy 3. Genealiie and Field Exeion Mahing he Magnei Field The Exiaion Coeffiien (PEC The Exiaion Coeffiien in ae of finie loe The Equaion Syem 46.6 The ongiudinal Couling medane 48.7 Numeial eul 55 Chae 3: The Thi i Genealiy and Field Exeion Mahing he Elei Field

4 Summay 3.3 The Exiaion Coeffiien (PECPMC The Equaion Syem The longiudinal ouling imedane Numeial Reul Chae 4: The Coaxial Seu Genealiie and Field Exeion Mahing he Magnei Field The Exiaion Coeffiien (PEC The exiaion oeffiien (oy The ongiudinal Couling medane Numeial eul... 8 Chae 5 : Comaion among Reul of aiou Code Genealiie Analiy of he Pillox ong. Couling medane Comaion wih he Reul of Wie Mehod Comaion of he eul wih a ommeial 5 Conluion 53 Aendix A Aendix B

5 Summay Aendix C Aendix D Biliogahy

6 Summay 6

7 noduion noduion aely we have aied o a damai ineae of he deign and ealiaion of high enegy and high ineniy aeleao. n hi one ae he high ineniy ina fo oduion of ulaviole and X ay. Thee aiviie imulaed eveal moan Comanie (i.e. Miuihi o odue inumenaion eialied fo he geneaion and he aeleaion of eleon eam of high ineniy, whee hee eam oni in vey ho unh ain (u o ome en of nanomee. One of mo imoan feaue of hi inumenaion i o avoid he eam qualiy euaion eaue of he ineaion wih uounding medium. Thi ineaion aea eaue he unhe dag image uen and, onequenly, Eleomagnei Enegy i oed in he uounding medium. A defomaion of he EM Enegy deniy may odue inene EM Field aing on he imay hage. Thi henomenon i eeened y he definiion of ad ho aamee: hey ae he Mahine medane and he Wae Field elevan o he fequeny domain and ime domain eeively. f hee aamee exeed ome hehold, we have a deeioaion of he eam qualiy and moe in geneal, limi he maximum eam enegy availale. A fi aoah o hi olem i o olve he EM field equaion wih he aoiae ounday ondiion and oue y mean of numeial ode. aiou ye of he aove menioned ode have een develoed (ABC, ROCOCO, 7

8 noduion Paile Sudio, e. whih hould give he Couling medane (Wae Field a funion of fequeny, fo a vaiey of onfiguaion of he uounding medium. Thee ae indiaion ha, a lea fo ome onfiguaion and ounday ondiion, he eul of hee ode ae no aifaoy. Maye hi i due o he equied muli-uoe goal, whih aifie he eliailiy o he veailiy. n ome ae, he eul even violae ome fundamenal hyial inile a Enegy Conevaion. We will analye hi ehaviou and we will fomulae ome hyohei on he aue of hee violaion. The imoiiliy o umi he oion of he mahine o e eed uing in loo a aile eam, in ode o oain he aamee of inee, foe he eeahe o limi hei e and exeimen o enh meauemen on he Devie Unde Te (DUT in a woho. n hi ae, a ommon aoah i o eo o an exeimenal ehnique, inodued in 974 y M.Sand e J.Ree on inuiive ai. Thi ehnique oni in elaing he unh y an imule iding on a wie, and o efom meauemen y mean of a Newo Analye. f he TEM field odued y he elei imule faily eodue he EM field geneaed y he unh, he EM ehaviou of he DUT indued y he wie, may give good indiaion on he ehaviou of he DUT ineaing wih he aile eam. Thi mehod, whih may give enile indiaion, i ill lagely adoed fo eing omonen of aile aeleao. Howeve, even if hi mehod ha een lagely udied, i eul ae only aially eliale a lea in ome ange of fequenie, a i will e hown. 8

9 noduion Thi i due o he eene of he wie ha may eu he meauemen maing uneain ome eul. We inen o eo o analyi/numeial mehod o e imlemened in homemade ode, oneived ad ho, and heefoe moe eliale han muliuoe ommeial ode. The ooed mehod ininially limi hei aliaion o he analyi of anonial model of ahe imle onfiguaion: ylindial ineion ouled o ylindial vauum an uh a, aviie, hi iie e. To hi end he mehod of Mode Mahing (MM ha een adoed. The ai idea of hi ehnique i o eeen he EM field in he aviie and in he waveguide y mean of a omlee e of ohogonal eigenmode (veo of hee iem, onideed a iolaed and wih homogeneou ounday ondiion. n aie, hi eeenaion i ueful eaue a limied nume of mode ae in geneal uffiien o have a good eeenaion of he field ehaviou and heefoe of he eleomagnei ineaion eween he aile eam and he uounding medium. When alied o finie domain, he Mode Mahing Tehnique mae eay o inodue ohmi loe, in ode o emulae eal uue. The uue o analye i divided in ue in whih i i oile o idenify aionay o avelling mode. n he ue of infinie dimenion, only he avelling wave ae aen ino aoun. Geneal ule o udivide in ue he devie do no exi: one mu oeed y aem. n ode o olve he olem, one ha o find he unnown oeffiien of he modal exanion. On he ode of adjaen ue, one ha o imoe he 9

10 noduion oninuiy ondiion of he EM-field. Thu, a e of funional equaion ae oained, equal o he ode ufae. Adoing he Ri-Galein mehod, one an hen oje hee equaion on an ohogonal funion e and, heefoe, hey hange ino an equal nume of maix/veo equaion. n he fi hae he eeah wa oiened on he udy of an ideal ylindial illox (PEC-PMC aviy and one wih finie onduiviy, oh ineed ino a ylindial vauum hame. n he ideal ae, onideaion aou enegei alane enfoe he eal a of he imedane o e ily eo elow he uoff fequeny even if in hi ange eonane exi. The eond e in he wo wa o veify he ageemen of he eul y MM wih hoe oained y he Sehed Wie Mehod. The heoy develoed in onneion wih hi mehod give he longiudinal ouling imedane fom he aeing aamee odued y he meauemen on DUT y mean of a Newo Analye. Howeve, ine he MM an model he onfiguaion of he Sehed Wie Mehod, we have fi oheed he eul of a enh meauemen and he one oained y mean of he imulaion of he Wie Mehod y mean of MM. Thi wo, udivided in five Chae, an e ummaied a follow: Chae : in hi hae i i inodued he mo elevan aamee o analye he ineaion eween he eam and he uounding medium. Fuhemoe, hee i a deailed exlanaion of he adoed mehod.

11 noduion Chae : hee i diued he Mode Mahing Tehnique alied o a Pillox Caviy. We will ue homogeneou ounday ondiion and we will inodue maeial loe o imulae eal devie. Chae 3: n hi hae, he MM ehnique i alied o a Thi i. Thi i a ai udy of he eliailiy of he MM ehnique wih mixed ounday ondiion. A onvegene udy and a omaion wih anohe Mode Mahing onfiguaion will e exoed. Chae 4: Hee i an exhauive eamen of he mode mahing ehnique alied o Wie Mehod meauemen. Chae 5: n hi hae, we will omae he eul y ou adoed mehod wih hoe oained y ommeial ode. Fuhemoe, we will ovide exhauive eamen aou he eliailiy of he wie mehod fo fequeny ange ha inlude he fequeny egion elow he waveguide uoff.

12 noduion

13 Chae : Main aamee and adoed Mehod Chae : Main Paamee and adoed Mehod. Main aamee The way o oeae of an aeleao ongly ide on he eleomagnei ineaion exiing eween he unh of haged aile and he vauum hame in whih i oagae. The deailed nowledge of hi oe i neeay o imove he aeleao efomane. We an onide he eam a a e of haged aile unhe laed a a oe diane, whih hould eeve eeive ae-ime oiion aigned duing he eviou aeleaion oe. Tavelling inide he vauum hame, he eam indue a eonday eleomagnei field ha may influene i dynami. Fo a elaivii aile in a efe and homogeneou uue, he final effe of he eonday field i null. An aeleao an e een a a devie wih feeda yem in whih evey longiudinal o anvee eam euaion an e amlified o aenuaed y eleomoive foe eaed y he euaion ielf. The eleomagnei field 3

14 Chae : Main aamee and adoed Mehod indued y he eam i alled Wae Field eaue i emain ehind he moving hage. The udy of longiudinal and anvee dynami need he nowledge of ome fundamenal aamee: - he longiudinal wae oenial: i i he volage vaiaion of a hage, due o he field geneaed y anohe hage whih eede i; he longiudinal wae-field i eonile fo he enegy lo eaue i i eenially in hae wih he aile; - The anvee wae oenial: i ae ino aoun he anvee foe alied o he eam due o he anvee wae-field; i effe i o defle he eam and, a a onequene, i an geneae unh ehing. Thee ae ohe aamee exloied o haaeie he ineaion eamaeleao: The Wae Poenial fo hage uni ae alled Wae Funion. The Fouie Tanfom of he Wae Poenial i alled Couling medane and i eul a funion of he fequeny. The wo aamee ju inodued eeen wo diffeen deiion of he ame henomenon, he eleomagnei ouling eween eam and aeleao uue. They deend y uue hae and no y unh oeie. The wae oenial, mainly ued fo linea aeleao, allow a deiion in ime domain, while 4

15 Chae : Main aamee and adoed Mehod Couling medane eeen he olem in he fequeny domain. Uually, i i emloyed fo iula aeleao, fo hei inini eiodiiy. To ee define he one of wae field, we an onide fi a iuaion of a ingula aile ha will e ue o define a oin oenial wae funion. The final wae field will e he aveage value on he whole ineaion one fo evey eam aile. Theefoe, le u onide Fig. -, wih q a a moving hage wih fixed veloiy aallel o he vauum hame axi, a he veo ha indiae he anvee oiion, a he veo ha indiae he longiudinal oiion. q q Z Z Fig. -. Refeene Fame The eleomagnei field odued in he famewo y q an e oained eolving he Maxwell equaion wih aoiae ounday ondiion. Thi field influene he dynami of oh q and q. We an define he enegy lo y q a he wo ha he eleomagnei field doe on i []: 5

16 Chae : Main aamee and adoed Mehod ( F(,,,, d U v (. (, ; F(,,,, d U τ τ v wih F he oen Foe. We an noie ha eviou inegal ae alulaed on an infinie ah and i doen oeond o a hyi ondiion, u i i o undeline ha hee exeion ae an evaluaion of he enegy gain a good a he wave lengh i malle han he onideed lengh. Beide, we an onide he longiudinal wae funion a he enegy ge y he eonday hage q fo hage uni q and q []: (, ; U (, ; τ τ [/C] (. q q w The lo fao a he lo enegy y q fo uni of quaed hage ( ( U [/C] (.3 q 6

17 Chae : Main aamee and adoed Mehod Finally, we an define he ongiudinal Couling medane a he Fouie Tanfom of he wae funion fo a oin lie hage. Z j ( ω τ ωτ //, ; w (, ; e dτ (.4 The wae funion an e oained fom longiudinal medane hough he Fouie Ani-anfom. Beide, i oun he yial oeie of he Fouie Tanfom. Anohe fomulaion of he ongiudinal Couling medane an e deived y he eviou fomula onideing a a oue a eam whih ha a longiudinal inuoidal modulaion in he aile deniy. Allowing fo he field odued y hi eam ineaing wih he uounding medium we an deive he ongiudinal Couling medane a funion of he wave nume. Z ( E (,e // j β d We may onide alo wae field and imedane odued y highe ode oue: diole, muliole e. Thee oue will lead o he elevan waefield and imedane. We will limi ouelve o he longiudinal ae and, fom now on, he longiudinal imedane will e alled imedane ou ou and he u index 7

18 Chae : Main aamee and adoed Mehod will e doed. Ju o give an examle, we give a ieion fo longiudinal ailiy of a oaing eam in a iula aeleao Z n m F β γ η ( / e Whee n hamoni nume e elemenay hage oed uen momenum ead η liage m aile e ma F fom fao (eween and.6 Geneally, he imedane i a omlex funion and fo hi eaon an e li in eal and imaginay a. The eal a eul elaed o eam loe. A we old efoe, when he unh oe vaiou ineion wih vaiale o eion inalled in he vauum hame, i exie eonday field: ome of hem emain loalied aound he unh and ohe ae loalied in eonaing uue and ohe oagae in he vauum hame. Thi aeion an e demonaed in he ideal ae of an infinie lengh vauum hame, eeening he longiudinal omonen of he elei field a a avelling wave hough he hame axi dieion wih andom hae veloiy. Fo 8

19 Chae : Main aamee and adoed Mehod high owe devie, wae field indued y aile an ongly modify he diiuion of he aeleaing field. Fuhemoe, when he unh oe a aviy, i exie no only he fundamenal mode u alo he high ode mode. They an indue eam enegy loe, admiane deeioaion (eenially in he unh aea and inailiy henomena wih aile loe. Geneally, a a onequene of hee effe, one an have evee limiaion of maximum elei uen iulaing in he aeleao. One an eaonaly affim ha udie on he wae field, on ouling imedane a a funion of he fequeny and, moe geneally, on he ineaion eween unh and uounding media, ae vey imoan o eah high qualiy eam oming ou he aeleao. Theefoe, i hould e ongly eommended in ojeing age, if i i oile, o loo fo: - Develoing aviie wih a le a oile high ode mode (and wih vey lile Q fao wih fequenie no oiniden wih he fundamenal mode highe hamoni, wih he uoe of eduing he ouling eween eam and high ode mode and heefoe o minimie he enegy loe; - Teing devie devoed o he aenuaion of high ode mode exied y he eam o avoid hem o ua enegy oed in he aviy. n ome ae, wih highly ollimaed high enegy eam, a he Fee Eleon ae one, hi goal i vey had o eah. 9

20 Chae : Main aamee and adoed Mehod. The wie mehod Thi ehnique wa ooed in 974 y M. Sand and J. Ree whih, on inuiive onideaion, wih he uoe o meaue he enegy lo y an eleon unh iding hough a aile aeleao omonen o e, a a eonan aviy. Thi mehod, ha allow o ge meauemen uffiienly meaningful wihou need o ue he aile eam u imly wih he nomal equimen fo enh meauemen, i i ill oadly ued in he udy of aile aeleao omonen. The imulaion of he unh aage hough he devie unde e (DUT i ealied ineing inide he uue a mealli wie along he eam axi, in whih flow a uen imule having a aial hae imila o he unh []. Thi onfiguaion allow o gain he aeing aamee of he onideed uue a feed y wo oaxial waveguide, and heefoe alo he longiudinal ouling imedane. Fig. -. Reeenaion of a iula o eion illox and he wie ehed along eam axi.

21 Chae : Main aamee and adoed Mehod The ai idea oni of onide o e oile, wih he uoe of he enegy loe evaluaion y a aile unh aued y he non-unifomiy in a vauum hame, he uiuion of he uen imule odued y he eam, wih a uen imule having he ame emoal ehaviou, flowing hough a wie ehed along he eam axi. One an ee ha he elei hage aoiaed o a aile eam oing hough a genei vauum hame odue inide of i eleomagnei field, whih odue on he wall of he uue a hage diiuion and indued uen. Sehing a mealli wie along he aviy axi, and negleing he ouling effe wih he inide adial line, i mae he aviy imila o a oaxial anmiion line. i woh of noe ha he inodued euaion oally modifie he ounday ondiion of he yem. in fa, he eion of he fundamenal uue oained will have no he imly onneion oey. A nown, hi ha a a onequene he oiiliy o have TEM mode and all fequenie oagaing mode a a oluion of he Maxwell equaion. Nevehele, aying a uen imule having he ame emoal ehaviou of he one elaed o he eam on he onduo, i ha een hown ha, he TEM field odued y hi imule exaly eodue he field odued y he eam if iniial enegy i equal o ha of he unh, unle in he immediae oximiy of he wie. The inuiion ugge ha indeendenly y he wie eene, he field geneaed iniially y he uen imule i he ame of he one odued y he eam, ovided ha he wie dimenion do no eu he eleomagnei field

22 Chae : Main aamee and adoed Mehod exiing wihou he wie. Afe hi, he fi hage and uen indued on uue wall an e held equal in he wo ae. Thi mean ha in a vey fi momen he aviy doen' anowledge he ounday ondiion vaiaion. All affimed ill now, aed exluively on inuiive onideaion, lead o elieve ha if he unh duaion eul o e mall in omaion o he ime of elaxaion of he aviy wih he wie, hen he enegy lo y he imule ha iulae on he onduo, and lo a eleomagnei enegy, i will e nex o ha lo y he aile eam emulaed. Theefoe, he eleomagnei ehaviou of he aviy, wih he wie ineed, i ongly indiaive of he aenuaion uffeed fom he High Ode Mode and a he ame ime allow o undeand he ouling eween aviy and eam, hu o aaie he aviy lo fao in funion of he fequeny []..3 The modal exanion in a ylindial aviy The ai idea of he ooed analyial aoah i o udivide he yem in ue (aviie and he waveguide haaeied y homogeneou ounday ondiion and o exand he field a a ueoiion of he elevan eigenmode.

23 Chae : Main aamee and adoed Mehod Fig. -3. Reeenaion of a iula o eion illox udivided in ue. The oluion i found y mahing he exanion oluion on he o ha eaae he ue. Thi an e eaily done fo he angenial omonen of magnei field, while fo he Elei Field i no oile eaue i angenial omonen on o i eo y definiion. The exanion ha non-unifom onvegene on hee oundaie. Howeve, i will e hown ha i i oile o oveome hi inonveniene. The omlee e of eigenmode oni in divegenele mode lu ioaional mode. Taing ino aoun he iula ymmey of he oundaie and of he exiaion in aviy, he field an e exeed in em of a omlee e of ai funion in a ylindial fame (,, in he following way [ 3]: 3

24 Chae : Main aamee and adoed Mehod 4 n n n n n n n n n n n n g G h H f F e E (.5 whee he aove mode aify he following equaion ; ; n n n n n n n n n n n n n n g g f f e e h h h e φ φ (.6 The ounday ondiion ae homogeneou fo he angenial Elei Field on he ufae S and fo he angenial Magnei Field on he ufae S, whee S U S i he whole ufae. Fuhemoe, a uual, he mode ae ohonomal, o ha i: nm m n m n nm m n m n d g g d f f d h h d e e δ δ ( ( ( ( ( ( ( ( * * * * (.7

25 Chae : Main aamee and adoed Mehod 5 ( ( ( ( * * * * ˆ ˆ ˆ ˆ m S m S m m m m S m m S m m nds e H jz nds h E nds e H Z nds h E j Z (.8 The olem of he non-unifom onvegene i olved [3] eoing o he modal exiaion oeffiien (he uen o he enion ae dawn onideing he ouling of he aviy wih he guide. Uing he Maxwell equaion and exloiing modal ohonomaliy, afe ome aage one an eah he following elaionhi eween he equivalen oue and he exiaion oeffiien S j ds H E n j * ˆ ζ ζ (.9 whee ζ i he imedane of he medium ha fill he aviy and nˆ i he uni veo ougoing fom ufae aviy. n he oagaion egion, he oagaion onan i:

26 Chae : Main aamee and adoed Mehod a π (. i i woh o noie ha he exeed angenial field in eq.(.9 won' e he ame exeed y eq.(.5 eaue he no unifom onvegene of he eie on aviy ufae. The inegal in eq.(.5, a we an ee afewad, an e alulaed only on he ouling ufae wih he guide and aing fom i mode. The oedue ha eainly ome iial oin on ounday edge in whih, he field would e infinie. Bu hi i no a olem, if we onide inegal aamee ha mediae on ome loal diffiulie. The oeffiien in he exanion and hoe oeonden of he guide ae unnown and hey mu e dawn y he ondiion of oninuiy guide-aviy, a we will how in nex Chae. 6

27 Chae : Main aamee and adoed Mehod 7

28 Chae : The PillBox Caviy Chae : The Pillox aviy. Genealiie and Field Exeion n hi hae, we will deal wih he illox aviy ae. We wan o alulae he ongiudinal Couling medane uing he Mode Mahing ehnique a aleady uefully done fo he ii. Thi ehnique an eaily analye he ouling eween he ylindial aviy and he waveguide haaeied y iula ymmey ha eeen he vauum hame a hown in Fig.-. e u onide a haged aile iding he oiive dieion, along he ymmey axi of a Pefe Elei Conduo vauum hame. We aume he aile moving wih onan veloiy, even hough he vauum hame dioninuiie would imly lile veloiy hange. ie in ii ae, hi aoximaion doe no affe ou alulu. Similaly, a i wa done fo he ii, we aume ha he foing imay field i odued y he aial eum of he eviouly menioned oin lie aile q, iding on he axi a veloiy v β. 3

29 Chae : The PillBox Caviy S S d S 3 Fig. -. Sheme of a Pillox aviy: d waveguide adiu; aviy adiu; aviy lengh. i woh of noe ha we have TM mode, wih adial and longiudinal omonen of Elei Field and aimuhal omonen of Magnei Field, a follow: E jqζ κ πγ β (, K ( κ ( κ ( κg K ~ ( κg H ( g- ex( j β ng( /β E qζ κ π β (, K ( κ ( κ ( κg K ~ ( κg H ( g- ex( j β (. H ϕ qκ π (, K ( κ ( κ ( κg K ~ ( κg H ( g- ex( j β ng( β whee g (in he ie; g (in he aviy, κ βγ, q i he aile hage (n he following fomula we adoed q fo imliiy, and H ~ ( g- i he Heaviide funion. 3

30 Chae : The PillBox Caviy 33 A exeed in deail in Aendix B, eonan mode in a ylindial uue of genei adiu g ae eeened y he fomula ( ( ( q q q g q J g J α π Φ (. whee g q q α α q i he q h eo of he equaion ( α J. The EM Taveling Mode inide a waveguide of adiu g ae ( ( ( q q q g q J g J α π Φ (.3 Fo a PEC ylindial aviy of adiu and lengh he nomalied eigenmode ae [3] ( ( ( ( ( ( ( [ ] ( ϕ ε ϕ ˆ ( o ˆ, (, ( ˆ o ˆ in ˆ ˆ h h ε, e, e, e Φ Φ Φ (.4

31 Chae : The PillBox Caviy 34 whee ε i he Neumann ymol ( ε if, ε ele and π. A diffeene wih he eviou ae, fo he aviy we will eo o eigenmode of all PEC ufae. Thi imlie a hange in he funion ha deie he longiudinal ehavio. Fuhemoe, we will no need o ae ino aoun he divegenele mode. The exlii exeion of he field i given a an exanion of he eigenmode weighed wih he exanion oeffiien and in he aviie and in he waveguide eeively: ( ( ( ( ( ( ( ( ( ex, ex, ex, < Φ Φ Φ j Z Y H j E j Y j E ϕ (.5 ( ( ( ( ( ( H Z E jz E ( o, ( in, ( o,,,, < < Φ Φ Φ ε ε ε ϕ (.6

32 Chae : The PillBox Caviy 35 ( ( ( [ ] ( ( ( [ ] ( ( ( [ ] j Z Y H j E j Y j E < Φ Φ Φ ex, ex, ex, ϕ (.7 whee ( Z Y α ( i he waveguide adiu and i he index of he h eo. The oal field inide all egion i given y he ueoiion of he imay field in eq.(. and he field ju defined. Thi ueoiion will e he exeion ued in he nex aagah, fo he Field Coninuiy.. Mahing he magnei field We ale he olem in he ame way a done fo he ii. Namely auming on he ufae and on he o he imay field and imoe ha he mode mu anel hi imay field. We may only onide he oninuiy of he magnei field angenial omonen on he wo o onneing he waveguide and he

33 Chae : The PillBox Caviy 36 aviy, a aleady done fo he Elei Field in he ii ae. On he ufae, he oal magnei field oninuiy i wien a: ( ( ( ( [ ] ( ( ( ( ( [ ] ( H H H H H H H H H H ~,,,, ~,,,, ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ (.8 whee we have aen ino aoun he imay field indiaed y uei, and ( H ~ i he Heaviide e funion needed o limi he inegal ah o he o inead of he oal inefae ufae. We oninue following he ame oedue a done fo he ii ae, y adoing he Ri Galein mehod, ojeing eq.(.8 on he waveguide eigenfunion ( Φ and oaining he following yem: M Y Y A M Y Y A (.9 whee he veo A and A ae defined a

34 Chae : The PillBox Caviy A A ~ (, Φ ( ds [ H (, Φ ( ] H ( Hϕ S S A e j β ϕ ds (. and he maix M i defined a M ( Φ ( π Φ d (. and ε ( ε (. ae funion of he exiaion oeffiien. The exlii exeion of he veo A i and he maix M ae given in Aendix C. The yem exeion uing maie and veo i eeened in he following equaion 37

35 Chae : The PillBox Caviy A YY A YY M M T T (.3.3 The exiaion oeffiien (PEC The oedue adoed hee i analogou o he one adoed fo he ii again. Beaue of he new ounday ondiion, he only diffeene i ha we will eo o he exiaion oeffiien inead of. Theefoe, he exeion of i: jy ˆ S n E h * ds (.4 whee S define he ideal ylindial aviy ufae, E i he oal elei field, nˆ i he ougoing uniy veo ohogonal o hi ufae and h i given y eq.(.4. On he laeal ufae of hi ylinde, fo a lole maeial, nˆ E i null. e u define S and S a he ufae of he wo ae. On hee ufae, only he adial omonen of he oal Elei Field have o e aen ino aoun. Theefoe, one an wie he eq.(.4 a 38

36 Chae : The PillBox Caviy 39 ( ( ( ( ( ( ( (,,,,,, ˆ,, ˆ * S * S S * S * ds h E ds h E jy ds h E ds h E jy (.5 whee ẑ i he uni veo having he axi dieion. We imoe he ounday ondiion of he angenial Elei Field on he ufae S and S. Even if he adial omonen of he mode vanihe on hee ufae (ee eq.(.6, howeve hey mu ehave a non-unifom onvegene o value diffeen fom eo uh a o aify he following ondiion: ( ( ( ( ( ( ( ( ( ( H E E E H E E E Φ Φ ~,,, ~,,, (.6 Thi equaion fo he angenial omonen an e ineeed ha he Elei Field of he eonan mode lu he imeed Elei Field inide he aviy: Fo < < mu e eo eaue of he efe onduing wall on he oona.

37 Chae : The PillBox Caviy Mu e oninuou and equal o he um of he Elei Field of he aveling mode lu he imeed Elei Field inide he waveguide. Taing ino aoun he definiion of he maix elemen M given y eq.(., one may ge he fomula: ( N M N M jy ε ( (.7 whee he nown veo N and N, he exended exeion of whih i given in Aendix C, ae given y he following fomula: N N π ~ [ E (, H ( E (, ] Φ ( N e j β d (.8 By ineing he ju wien equaion in he exeion of and given y eq.(., we ge: 4

38 Chae : The PillBox Caviy 4 ( ( M N M N jy M N M N jy ε ε ε ε (.9 A ey feaue of hee exeion i ha he wo um wih he -index an e u in a loed fom. Thi i a geneal oey and i i elaed o he modal exanion of Geen Funion. n addiion o he undoued advanage of he analyial um, one ha he fuhe advanage ha he maie ae edued of one dimenion. The eleomagnei olem will alo enefi of hi ehavio: all he longiudinal eleomagnei mode ae aen ino aoun and heefoe only a few anvee mode ae uffiien o deie he henomenon. Fo he um of he eie in quae ae, we ado he ame oedue a done fo he ii, o ( ( ( ( N M Z N M Z jy Y N M Z N M Z Y jy o o (.

39 Chae : The PillBox Caviy whee Z ( α Y ( i he aviy o eion adiu. Exeing he aove fomula in em of maie and veo we ge: jy Y jy Y [ ( Z ( M N o( Z ( M N ] [ o( Z ( M N ( Z ( M N ] (..4 The Exiaion Coeffiien in ae of finie loe So fa, we have diued of a PEC devie. Theefoe, we oe aou a uue haaeied y he following oeie: Elei field i eendiula o he wall. The uen in he wall i a ufae uen. No enegy diiaion in he wall. Below uoff he field amliude eome infinie a eonane fequenie. When we onide a loy devie, he aove oeie will hange eaue of he finie onduiviy. Thee will e enegy loe in he wall ha limi he Field amliude a he eonane fequenie. will e ill vey lage, u no infinie. Fuhemoe, he uen in he wall eome a volume uen wih eneaion de deending on he maeial. The mo imoan hange fo ou evaluaion i ha he Elei field ha a angenial omonen a he wall. Thi omonen i 4

40 Chae : The PillBox Caviy vey lile and deend on he eneaion deh, u we need o add i o eq.(.4 a a em unde inegal. The finie onduiviy i eeened y he ufae imedane j Z S (. σ δ whee σ and δ eeen he maeial onduiviy and he eneaion de eeively [3, 4]. The Sufae medane ha he meaning ha, on he mealli ufae, hee i a elaionhi eween he angenial omonen of he Elei Field and he Magnei Field, given y: E l Z nˆ H (.3 S whee he ui l indiae loe. Thi em ha o e added o he eq.(.4 and hen he exiaion oeffiien fomula eome: 43

41 Chae : The PillBox Caviy 44 ( ( ( ( S vw vw vw S S S o S S S S o S S S l o d h h Z jy nd h E jy d h H Z jy nd h E jy d h E n nd h E jy (.4 We ema ha he exiaion oeffiien ae ignifianly diffeen fom eo only a fequenie uh ha Theefoe, in he aove equaion we may negle in he um all he exiaion oeffiien u. So ha, we may aoximae he exeion given y eq.(.4 a ( ( ( ( ( [ ] Φ S S S S o S S S S o ds Z jy nds h E jy ds h h Z jy nds h E jy * * * o ε (.5

42 Chae : The PillBox Caviy By mean of ome algea deied in he Aendix C, we finally oain he following fomula: ( jy jy Z S ε S S * ( Eo h nds (.6 Befoe uiuing he ju wien fomula in eq.(. o oain he exiaion oeffiien, we need o eo o a new definiion of modal imedane a: Z And define he new quaniie δ jy Z S [( ]( [ jy Z ] S j Y Z S (.7 A demonaed in Aendix C, he exiaion uen fo a loy Pillox ae: 45

43 Chae : The PillBox Caviy jy jy {[ δ Y ( Z ] ( M N [ δ Y o( Z ] ( M N } {[ δ Y o( Z ] ( M N [ δ Y ( Z ] ( M N } (.8.5 The Equaion Syem The exiaion uen exeed in eq.(. and in eq.(.8 allow u o eah he ulimae exeion of he eq.(.3 yem in eihe lo fee and loy ae. We will eo eihe fomula, even if i i oile o hif fom loy o lo fee ae imly equaing o eo he aamee defined in eq.(.7. Fo a lo fee illox, equaing eq.(.3 wih eq.(. we ge he following yem: A YY A YY jy M Y T jy M Y T [ ( Z ( M N o( Z ( M N ] [ o( Z ( M N ( Z ( M N ] (.9 By mean of ome algea i i oile o unoule he unnown and, heefoe, o imlify he oluion. By adding and uaing he wo exeion, we oain: 46

44 Chae : The PillBox Caviy T [ jz M Y o( Z M ]( Z Z T ( A A jz M Y o( Z ( N N (.3 T [ jz M Y an( Z M ]( Z Z T ( A A jz M Y an( Z ( N N (.3 whee i he ideniy maix. Hee we eoed o he following igonomei exeion: an ( x / ( x o( x and o( x / ( x o( x (.3 Fo a loy illox, equaing eq.(.3 wih eq.(.8 a aleady done fo he lo fee illox, we ge he following yem: A Y Y T jy M A YY T jy M {[ δ Y ( Z ] ( M N [ δ Y o( Z ] ( M N } {[ δ Y o( Z ] ( M N [ δ Y ( Z ] ( M N } (.33 47

45 Chae : The PillBox Caviy By adding and uaing he wo exeion and alying he igonomei exeion hown in eq.(.3, we oain: T { jz M [ δ Y o( Z ] M }( Z Z [ ]( N N T ( A A jz M δ Y o( Z (.34 T [ jz M Y an( Z M ]( Z Z T ( A A jz M Y an( Z ( N N (.35 A ommonly ued omue ool, Mahwo Mala, eaily olve he ju oained equaion. Aually, i i neeay o unae he infinie maie efoe ying o olve he equaion. n eion.7 we will how a good mehod o unae he maie wihou loe eul goodne..6 The ongiudinal Couling medane We deemine he Couling imedane eaaing he inegal in omonen elaed o he aviy egion whee, o e onien wih he eviou aumion, we ae he hage q. 48

46 Chae : The PillBox Caviy 49 d,e ( E d,e ( E d,e ( E Z( β j β j β j (.36 he longiudinal omonen of he Elei field fo he hee egion i given in eq.(. and ummaied a follow. ( ( ( ( ( ( ( ( [ ] j Y j E jz E j Y j E < Φ < < Φ < Φ ex, ( o, ex,, ε (.37 Fo he < inegal, uiuing he Elei field fo and deived of he fao no elevan fo inegaion, we oain an exeion lie he following d β j j ex (.38 whih i he ame of he inegal x x e dx e α α α

47 Chae : The PillBox Caviy 5 Reoing o he aove fomula o eolve he inegal in eq.(.38 we oain β β j ha an e wien in a moe elegan exeion (uing κ /βγ κ β j (.39 Reuning o he geneal exeion of he inegal, we noie ha ( ( ( q q q q ρ q α πj ρ α πj ρ J ( Φ Theefoe, uiuing eveyhing in he fi inegal of eq.(.36 we have ( ( β j κ α J β Y π d,e ( E (.4

48 Chae : The PillBox Caviy 5 Fo wha onen he inegal of ongiudinal medane elaed o he ae <<, he Elei field along he axi i given in eq.(.37. Realling he exlii exeion of ( M N M N jy ε we oain he exeion of he inegand ( ( ( ( Φ M N M N o ε (.4 Fo and wihou elemen no elevan o he inegal oluion, we oain he following inegal and he elaive oluion ( ( ex ex o j j d j β β β β (.4 Theefoe, he inegal of he ongiudinal Couling medane we ae looing fo, will e

49 Chae : The PillBox Caviy 5 ( ( [ ] ( ( ( ~ ~ ex, j e j J B A d j E β β α π ε β β (.43 whee M N B M N A ~ ~ The um on an e analyially alulaed a aleady een fo ii, wih a imila eul. n fa, fom he inegal eul we gain fou um on o olve, deived y he follow: ( β ε ( ( β ε Solving he aove um we eah hi eul

50 Chae : The PillBox Caviy 53 ( ( ( ( [ ] ( ( ( [ ] o ~ o ~ j j j e e j B e j A J β β β β β β α π β β β β (.44 Afe ome algea we oain he final eul fo he eond inegal of eq.(.36 ( ( ( ( ( ( ( β j β j β j α J κ Z e B A B e A πβ j α J κ e M π d j, E ~ ~ o ~ ~ ex β (.45 Fo he > inegal, uiuing he Elei field fo and wihou elemen no elevan o he inegal oluion, we oain an exeion lie he following

51 Chae : The PillBox Caviy 54 ( d j β j ex (.46 The aove inegal an e wien a aleady done fo he inegal of he fi egion ( d β j j ex ex (.47 whih ha he ame oluion, wih diffeen ign. Solving he inegal and uing κ /βγ we oain hi eul β j ex κ β j (.48 Theefoe, ealling he exeion of ( Φ q q ρ and uiuing eveyhing in he hid inegal of eq.(.36 ( ( β α π j J Y ex κ β d,e ( E β j (.49

52 Chae : The PillBox Caviy The ongiudinal Couling medane of he Pillox i given y he um of he hee inegal eq.(.9, eq.(.34, eq.( Numeial eul The wo Equaion Syem (eq and eq involve infinie equaion and infinie unnown. To allow he yem inveion i i neeay o unae he infinie maie wihou ha he eul validiy. A aleady een fo he ii, hough he Relaive Convegene henomenon, i i oile o eah a diffeen eul fo diffeen maix unaion. Following he heme eoed on ee and Mia oo [6] we imoed a elaion eween he nume of mode of diffeen one in ode o ee he Meixne ondiion [7]. The hoie of he aio N/N and N3/N (whee Ni indiae he nume of mode fo he i h egion ha a onideale effe on he eul goodne a een fo he ii. N N (.5 w n ou eifi ae, we hoen 55

53 Chae : The PillBox Caviy N N N 3 N / w / w / w N / w / w N / w3 / w / w (.5 whee N N N N3 and w 3 w (eaue d, hen N N3. Afe unaion and inveion of he linea equaion, we olved he olem. will e eeened he ongiudinal Couling medane, a a fundamenal aamee fo aeleao oje, udivided in eal and imaginay a and fo diffeen value of nume of mode, geomeial aamee and aile eed. The nume of oin i hoen a a imulaion onan, n 5, and he ame i done fo he waveguide adiu ( mm. The nume of mode i fixed (N, u in ome ae hi nume may e hanged, when i i needed o ineae i o eah he onveegene, a aleady diued fo ii ae. 56

54 Chae : The PillBox Caviy Fig. -. ongiudinal Couling medane, eal a: βγ, / 4, / 4. Fig. -3. ongiudinal Couling medane wih loe, eal a: βγ, / 4, / 4, ρ/(

55 Chae : The PillBox Caviy Fig. -4. ongiudinal Couling medane, imaginay a: βγ, / 4, / 4. Fig. -5. ongiudinal Couling medane, eal a: βγ, / 6, / 4. 58

56 Chae : The PillBox Caviy Fig. -6. ongiudinal Couling medane wih loe, eal a: βγ, / 6, / 4, ρ/( Fig. -7. ongiudinal Couling medane, imaginay a: βγ, / 6, / 4. 59

57 Chae : The PillBox Caviy Fig. -8. ongiudinal Couling medane, eal a: βγ, / 8, / 4. Fig. -9. ongiudinal Couling medane wih loe, eal a: βγ, / 8, / 4, ρ/(

58 Chae : The PillBox Caviy Fig. -.ongiudinal Couling medane, imaginay a: βγ, / 8, / 4. Fig. -.ongiudinal Couling medane, eal a: βγ, / 4, / 4. 6

59 Chae : The PillBox Caviy Fig. -. ongiudinal Couling medane wih loe, eal a: βγ, / 4, / 4, ρ/( Fig. -3.ongiudinal Couling medane, imaginay a: βγ, / 4, / 4. 6

60 Chae : The PillBox Caviy Fig. -4.ongiudinal Couling medane, eal a: βγ, / 6, / 4. Fig. -5. ongiudinal Couling medane wih loe, eal a: βγ, / 6, / 4, ρ/(

61 Chae : The PillBox Caviy Fig. -6.ongiudinal Couling medane, imaginay a: βγ, / 6, / 4. Fig. -7.ongiudinal Couling medane, eal a: βγ, / 8, / 4. 64

62 Chae : The PillBox Caviy Fig. -8. ongiudinal Couling medane wih loe, eal a: βγ, / 8, / 4, ρ/( Fig. -9.ongiudinal Couling medane, imaginay a: βγ, / 8, / 4. 65

63 Chae : The PillBox Caviy Fig. -.ongiudinal Couling medane, eal a: βγ, / 4, / 4. Fig. -. ongiudinal Couling medane wih loe, eal a: βγ, / 4, / 4, ρ/(

64 Chae : The PillBox Caviy Fig. -.ongiudinal Couling medane, imaginay a: βγ, / 4, / 4. Fig. -3.ongiudinal Couling medane, eal a: βγ, / 6, / 4. 67

65 Chae : The PillBox Caviy Fig. -4. ongiudinal Couling medane wih loe, eal a: βγ, / 6, / 4, ρ/( Fig. -5. ongiudinal Couling medane, imaginay a: βγ, / 6, / 4. 68

66 Chae : The PillBox Caviy Fig. -6. ongiudinal Couling medane, eal a: βγ, / 8, / 4. Fig. -7. ongiudinal Couling medane wih loe, eal a: βγ, / 8, / 4, ρ/(

67 Chae : The PillBox Caviy Fig. -8.ongiudinal Couling medane, imaginay a: βγ, / 8, / 4. Fig. -9.ongiudinal Couling medane, eal a: βγ, / 4, / 4. 7

68 Chae : The PillBox Caviy Fig. -3. ongiudinal Couling medane wih loe, eal a: βγ, / 4, / 4, ρ/( Fig. -3.ongiudinal Couling medane, imaginay a: βγ, / 4, / 4. 7

69 Chae : The PillBox Caviy Fig. -3. ongiudinal Couling medane, eal a: βγ, / 6, / 4. Fig ongiudinal Couling medane wih loe, eal a: βγ, / 4, / 6, ρ/(

70 Chae : The PillBox Caviy Fig. -34.ongiudinal Couling medane, imaginay a: βγ, / 6, / 4. Fig ongiudinal Couling medane, eal a: βγ, / 8, / 4. 73

71 Chae : The PillBox Caviy Fig ongiudinal Couling medane wih loe, eal a: βγ, / 4, / 8, ρ/( Fig. -37.ongiudinal Couling medane, imaginay a: βγ, / 8, / 4. 74

72 Chae : The PillBox Caviy When he value of he eal a of he Couling medane ae a low a Ohm, i i uele o ineae he nume of mode: he eul ae quie eai. Thi i a onequene ha we ae aoahing o he auay given fo he eo of he Beel Funion [8]. Thi inonveniene, whih haen a vey low enegie, affe only he eal a of he Couling medane and no he imaginay a. Fig Convegene e fo Mode Mahing Tehnique, Couling medane. (βγ., /, /4. 75

73 Chae : The PillBox Caviy 76

74 Chae : The Thi i Chae 3: The Thi i 3. Genealiy and Field Exeion Hee we wan o analye he ineaion eween a eam and a hi ii ineed in a ylindial vauum hame oh of iula o eion. S S d v β ZONE ZONE ZONE Fig. 3-. Shemai eeenaion of a genei Thi i. We aume he foing imay field a odued y he aial eum of a oin lie aile q iding on he axi wih a veloiy β. Theefoe, he imay field ae TM ye and hei exlii exeion ae given elow 77

75 Chae : The Thi i E jqz κ πγ β (, K ( κ ( κ ( κg K ( κg ex( j β ng( /β E qz κ π β (, K ( κ ( κ ( κg K ( κg ex( j β (3. H ϕ qκ π (, K ( κ ( κ ( κg K ( κg ex( j β ng( β whee g (in he waveguide; g (in he ii, κ βγ, K m and m ae modified Beel Funion, (n he following fomula we adoed q fo imliiy. n Aendix A i given a deailed exoiion. Thi onfiguaion i li in wo emi-infinie ie, eaaed y a ylindial egion of he ame adiu a he ii and of ame lengh. The ie ae onideed a waveguide and he ylindial egion a a illox aviy. The ai idea of he analyial aoah i o eeen he EM Field in he aviy and in he waveguide y mean of eigenmode of hee uue a onideed iolaed and wih efe (magnei o elei wall. i well nown ha hee mode fom a omlee e y mean of whih we an eeen any EM Field onfiguaion. Then, in ode o find he exanion oeffiien, we mu imoe he oninuiy of he elei and he magnei field on he ode eaaing adjaen uue. The ounday ondiion on he ii ae: Pefe magnei onduo on he ae Pefe elei onduo on he laeal ufae 78

76 Chae : The Thi i. m.. nˆ.e... m...e.. Fig. 3-. Sheme of he ounday ondiion. nˆ The imay field ae of TM ye. The ylindial ymmey doe no inodue any longiudinal Magnei Field. Theefoe, he aeed field will e of TM ye oo. One an ale he olem in wo diffeen way:. Aume he imay oue and imoe he ounday ondiion on he ufae and he oninuiy on he o [4]. Aume on he ufae and on he o he imay field and imoe ha he mode mu anel hee imay field. We will ado he eond aoah. i lea ha he imay field alone doe no aify all he ounday ondiion: fo inane, he angen elei imay field on he oona i no vanihing. Theefoe, he mode mu eoe hi ondiion on hi ufae. The EM Tavelling Mode inide a genei ylindial uue of adiu g an e eeened y he following nomalied eigenmode: 79

77 Chae : The Thi i J ( q g Φ q ( q (3. g π J ( α q J( q g Φ q ( q (3.3 g π J ( α q whee q α g and α q i he q h eo of he equaion J ( α. Moe deail q aou hee exeion ae wien in Aendix B. Fo a ylindial aviy of adiu and lengh he fomula of he nomalied eigenmode i wien [4] a e (, e (, ˆ e (, ˆ ε [ o( Φ ( ˆ in( Φ ( ˆ ] (3.4 h (, h (, ˆ ϕ ε in ( Φ ( ˆ ϕ whee ε i he Neumann ymol ( ε if, ε ele and π. The field inide he hee one in whih we divided he devie ae eeened a follow 8

78 Chae : The Thi i 8 ( ( ( ( ( ( Φ Φ Φ j j j e Z Y H e E e Y j E,,, ϕ (3.5 ( ( ( ( ( ( Φ Φ Φ H F E F E,,, ( in, ( o, ( in, ε ε ε ϕ (3.6 ( ( ( ( ( ( ( ( ( Φ Φ Φ j j j e Z Y H e E e Y j E,,, ϕ (3.7 whee j j Z ωµ ; ; ε i he Neumann ymol ( ε if, ε ele ; ( Z Y α ( i he ie o eion adiu; and F ae he exiaion oeffiien of he divegenele and ioaional eonan mode eeively.

79 Chae : The Thi i The oal field inide evey egion i given y he ueoiion of he imay field in eq.(3. and he field ju defined. Thi ueoiion will e he exeion ued in he nex aagah, fo he Field Coninuiy veifiaion. 3. Mahing he elei field Now we will define he Elei field oninuiy a he inefae eween he hee ue in whih we divided he devie, aing ino aoun he ioaional mode oo: E E ~ (, E (, [ E (, E (, ] H ( ~ (, E (, [ E (, E (, ] H ( (3.8 whee he imay field ae indiaed y he uei, and H ( ~ i he Heaviide e funion ha eeen he negleing of he field in oeondene o he oona. Fom eq.(3.8 and uing he exeion of he Tanvee Magnei Mode, we ge he exlii exeion of he oninuiy a he inefae: 8

80 Chae : The Thi i 83 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Φ Φ Φ Φ H F e K K Z e K K Z H F K K Z K K Z j j ~ ~ ε κ κ κ κ π β κ κ κ κ κ π β κ ε κ κ κ κ π β κ κ κ κ κ π β κ β β (3.9 whee and ae he waveguide and aviy adii eeively; indie, indiae he lef and he igh inefae eween ue. By adoing he Ri Galein mehod, we oje eq.(3.9 on he eigenfunion ( Φ and we oain he following yem: M A M A (3. whee he veo A and A ae elaed o he imay field and ae defined a

81 Chae : The Thi i 84 ( ( ( ( ( β j S S e A A H ds E ds E A ~,, Φ Φ (3. whee S indiae he inefae ufae and he maix M i defined a ( ( Φ Φ d M π (3. and hei exlii exeion i given in Aendix C. i woh of noe ha i i needed o ge he um e defined a: ( F F ε ε (3.3 The ju given definiion will inodue a imlifiaion eaue one index ha een auaed.

82 Chae : The Thi i 3.3 The exiaion oeffiien (PECPMC The mode exiaion oeffiien and F defined in he aove aagah, ae deemined aing ino aoun he ouling eween aviy and waveguide. Uing Maxwell Equaion and he modal ohonomaliy, afe ome algea i i oile o eah he fomula we ae looing fo. Oeaing in uh way, i i imlii o imoe he oninuiy of he angenial Magnei Field on he inefae eween aviie and waveguide. Thi mean ha we uild a non-eo Field in oeondene of he aviy o, uing Field diiuion ha eul eo on he ame o (aviy mode. Thi oeaion i done eaue he um onvege no unifomly, o he limi of he um alulaed in a oin ha lie on he efe magnei onduo ufae anno e hanged o he um in a oin whoe limi end o he ufae of he onduo. n fa, he fi limi i eo, wheea he eond end o he aigned Magnei Field. Thi oedue ha a iial oin in oeondene of he edge angle whee he field hould end o infinie. Howeve, hi effe i no a eal olem eaue he aamee we ae alulaing ae of gloal ye, o hey mean on hee iial oin. Summaiing wha wien aove, he oninuiy of he Elei Field anno e exeed ou ou uing he eigenfunion exanion eaue hey aify he homogeneou ounday ondiion. Howeve, hi inonvenien an e iumvened eoing o he exiaion oeffiien a funion of he Magnei Field diiuion on he o [3] a: 85

83 Chae : The Thi i jz * ( H e S nˆ ds (3.4 whee nˆ i he ougoing uniy veo ohogonal o he aviy ufae S and e i given y eq.(3.4. We will have a non-eo Elei Field angen omonen only on he wo o S and S. Theefoe, one an wie he eq.(3.4 a: jz jz S H S ϕ ˆ H e * * (, ds ˆ H e (, * * (, e (, ds Hϕ (, e (, S S ds ds (3.5 whee ẑ indiae he oiive dieion of axi fom lef o igh and H eeen he oal Magnei Field a he inefae ufae. We imoe he ounday ondiion of he angenial Elei Field on he ufae S and S. Even if he adial omonen of he mode vanihe on hee ufae (ee eq.(3.8, howeve hey mu ehave a non-unifom onvegene o value diffeen fom eo uh a o aify he following ondiion: 86

84 Chae : The Thi i 87 ( ( ( ( [ ] ( ( ( ( ( [ ] ( H H H Z Z H H H H Z Z H Φ Φ ~,,, ~,,, ϕ ϕ ϕ ϕ ϕ ϕ (3.6 Exanding he inegal and aing ino aoun he definiion of he maix elemen M one may ge he fomula: ( ( Z Z M N Z Z M N Z j ε π (3.7 whee ( ( [ ] ( β ϕ ϕ π j e N N d H H N,, Φ (3.8 and i exlii exeion i given in Aendix C. Fo wha onen he ioaional mode we have [4]

85 Chae : The Thi i F jβ π S ( f H nˆ ds (3.9 whee n i he ougoing veo ohogonal o he aviy ufae S and f i given y f (, ε ε o o ( J ( π J ( α ( ( ( ( ( J ˆ in J ˆ π J α (3. We will have a non-eo Elei Field angen omonen only on he wo o S and S. Theefoe, one an wie F ( f (, H nˆ ds f (, jβ π S S ( H nˆ ds (3. whee H eeen he oal field a he inefae ufae. Exanding f we oain afe ome algea F jπ Z ε ( H ( H Φ ( d ϕ ϕ (3. 88

86 Chae : The Thi i 89 Taing ino aoun he definiion of and a given y eq.(3.3, we ge: ( ( ( ( ( ( Z M N Z Z M N Z j Z M N Z Z M N Z j ε ε (3.3 A ey feaue of hi exeion i he wo um wih he -index an e u in a loed fom. Thi i a geneal oey and i elaed wih he modal exanion of Geen Funion. Reoing o eq.(.4.4 of efeene [5] one an oma he um ove a: ( ( o ε (3.4 ( ( ( ε n addiion o he undoued advanage of he analyial um, one ha he fuhe advanage ha he maie ae edued of one dimenion. The eleomagnei

87 Chae : The Thi i olem will alo enefi of hi ehaviou: all he longiudinal eleomagnei mode ae aing ino aoun and heefoe only a few anvee mode ae uffiien o deie he henomenon. Fom he aove algea, we oain hen he imlified fom: jz jz o ( Z Z N M ( Z ( Z Z N M o( Z Z Z ZN ZN M M Z Z (3.5 whee Z ( α Y ( i he ii o eion adiu. Exeing he aove fomula in em of maie and veo, we ge: jz [ ( Z ( Z N M Y o( Z ( Z N MY ] jz [ o( Z ( Z N M Y ( Z ( Z N MY ] (3.6 whee Z and Y ae diagonal maie. 9

88 Chae : The Thi i 3.4 The Equaion Syem By equaing eq. (3. wih eq. (3.6, we ge he following infinie yem in infinie unnown: A A jm jm T T Z Z [ ( Z ( ZN M Y o( Z ( ZN MY ] [ o( Z ( Z N M Y ( Z ( Z N MY ] (3.7 By mean of ome algea i i oile o unoule he unnown and, heefoe, o imlify he oluion. By adding and uaing he wo exeion, we oain: T [ jm Z o( Z MY ]( T ( A A jz M Z o( Z ( N N (3.8 T [ jm Z an( Z MY ]( T ( A A jz M Z an( Z ( N N (3.9 Whee we eoed o he following igonomei exeion: 9

89 Chae : The Thi i 9 ( o( ( / an x x x and ( o( ( / o x x x A a onluion one an ee ha he equaion ae unouled ine in he fi one aea only he unnown ( and in he eond one only (. Theefoe, hey an e olved y he inveion of a imle maix. Ohe auho in a imila way olve he olem fo hi ae eoing o a wave eeenaion inide he ii (Tavelling Wave Mode Mahing: ( ( ( ( ( ( ( ( ( Φ Φ Φ j j j j j j e e Z H e e E e e j E,,, π β π β π β ϕ (3.3 whee / α and α ae he eo of he Beel funion J (x and ( Φ and ( Φ ae he modal funion. Howeve, hei eul ae eied o he lole ae, u i i no he only limiaion. One an ee ha, eing equivalen o eah ohe, he mode-mahing ehnique need only half mode in ee o he nume of mode needed y avelling wave

90 Chae : The Thi i mode mahing. Thi mean ha ou ehnique need le omuaional owe o eah he eul han he laial avelling wave mode mahing. Howeve, hi advanage i no enough o juify he ineaed mahemai diffiulie inodued y hi mehod, due o no unifomly onvegen eie. A will e hown, he mixed mode mahing ehnique allow eahing ee eul han he ohe mehod. 3.5 The longiudinal ouling imedane When a uue a he ii i udied in an aeleao oje, i i imoan o evaluae i ieaion wih he aile eam. n ime domain, a gloal aamee ha define hi ineaion i he wae oenial, aleady defined. analogou in he fequeny domain i he longiudinal ouling imedane, eaily oained fom he oenial uing he Fouie Tanfom. Hee we will a fom he moe geneal definiion of he medane aleady given in eviou aagah, o eah a aiula exeion fied o he ii uue, a: Z( q E (, e j β d q q E E 3 (,e (, e j β j β d d. (3.3 93

91 Chae : The Thi i The oedue adoed i he ame a he one of he eviou Chae, and we ge he numeial eul lied in he nex Seion. 3.6 Numeial Reul The wo Equaion Syem (eq involve infinie equaion and infinie unnown. To allow he yem inveion i i neeay o unae he infinie maie wihou ha he eul validiy. Doing ome imulaion on he devie i oile o ee a diffeen eul fo diffeen maix unaion. i he Relaive Convegene henomenon [6]. A hi, a finie nume of mode fo eah waveguide and aviy mu e onideed. The hoie of he aio N/N and N3/N (whee Ni indiae he nume of mode fo he i h egion ha a onideale effe on he eul goodne. Following he heme eoed on ee and Mia oo [6] we imoed he wien elow elaion eween he nume of mode of diffeen one in ode o ee he Meixne ondiion [7]. N N (3.3 w n ou eifi ae, we hoen 94

92 Chae : The Thi i N N N 3 N / w / w / w N / w / w N / w3 / w / w (3.33 whee N N N N3 and w 3 w (eaue d, hen N N3. Afe unaion and inveion of he linea equaion, we olved he olem. will e hown he goodne of he mode-mahing analyi o manage he aile aing hough a hi ii olem. will e eeened he ongiudinal Couling medane, a a fundamenal aamee fo aeleao oje, udivided in eal and imaginay a and fo diffeen value of nume of mode, geomeial aamee and aile veloiy. 95

93 Chae : The Thi i Fig Comaion eween mixed mode mahing and avelling wave mode mahing alied on he ame devie: Real a of Couling medane (βγ, /., /.5 Fig Comaion eween mixed mode mahing and avelling wave mode mahing alied on he ame devie: maginay a of Couling medane. (βγ, /., /.5 96

94 Chae : The Thi i n Fig.3-3 i hown he omaion eween he mehod adoed hee and he avelling wave mode mahing (uually alled mode mahing in lieaue, alied o he ame uue. i woh of noe ha he nomaliaion of wave nume o he guide adiu imlie ha he uoff fequeny will alway fall on he ame value of he nomalied wave nume. Thi value (.4 oeond o he fi eo of he Beel Funion J (x. One hould no e uied y he vanihing of he eal a of he medane fo all he fequenie elow he uoff. A diffeen ehaviou would onfli wih he enegy onevaion inile. Allowing fo he enegy eleaed y he eam ino he oom delimied y he dioninuiy of he ii, hi enegy mu e eniely given a again o he eam ielf. Sine we ae elow he uoff, no enegy i indeed allowed o feely flow inide he ie. Theefoe, he eal a of Couling medane mu e eo eaue he eam did no loe any enegy. By onvee, he imaginay a i eainly diffeen fom eo ine hee i a alaned exhange of enegy eween he eam and he oom inide he dioninuiy, a hown in Fig.3-4. We exe ha hi will no haen when he wall of he ii have a finie onduiviy, ine a eain amoun of he enegy exhanged will e diiaed on he wall. We exe ha in hi ae a ome fequenie (elaed o he devie eonane i will aea a non-vanihing eal a in he Couling medane. Aove uoff, he eal a he ouling may e diffeen fom eo: a eain amoun of he enegy, eleaed y he eam ino he oom delimied y he dioninuiy of he ii, may flow ino he 97

95 Chae : The Thi i eam ie. Sine he hae veloiy of i EM field i lage han he aile veloiy, he mean owe exhange eween he eam and he field i eo: in um, hi enegy i ieveily lo and a non-eo eal a aea in he Couling imedane, even in he ae of lole wall. While a low fequeny he eul of he wo mehod ae almo ueoale, he dieany eween hee eul eome maooi a high fequeny, whee i aain almo %. n ode o undeand whih mehod i moe onvenien, we efomed a onvegene e fo he mixed mode mahing ae. The eul of hi oedue ae eeened in Fig.3-5. n Fig.3-3 i alo eoed he ime needed y he omue o eah he eul and i i lea ha in hi ae he avelling wave mode mahing mehod fae han Mixed mode mahing mehod. i only a oinidene, eaue uually he M.M. i ininially fae and, a moe lile i βγ a moe eviden i he diffeene in ime. 98

96 Chae : The Thi i Fig Fou Convegene e fo Mixed mode mahing Tehnique, maginay Pa of Couling medane. (βγ, /., /.5. Fig Convegene e fo Mixed mode mahing Tehnique, maginay Pa of Couling medane. (βγ, /., /.5. 99

97 Chae : The Thi i Thi mehod exhii a fa onvegene in he eimaion of he longiudinal ouling imedane. Few mode ae equied o oain an eo lowe han een and hen an auae value of he imedane. n Fig.3-5 and Fig.3-6 i hown he onvegene udy on he eal a of he Couling medane. f mode ae ued, one an e find an eo lowe han. fig Convegene of he eal a of he ongiudinal medane. (3. n he figue elow, we wan o how how muh he ongiudinal Couling medane i ueful o undeand he ineaion eween he aile and he ii fo ome aile eed value and vaying ome ii aamee. Evey imulaion

98 Chae : The Thi i i done uing mode and aing 5 oin fo evey gahi. The only onan aamee i he waveguide adiu 3 mm. Fig ongiudinal Couling medane, eal a: βγ., /., /.5.

99 Chae : The Thi i Fig ongiudinal Couling medane, imaginay a: βγ., /., /.5. Fig. 3-. ongiudinal Couling medane, eal a: βγ., /.4, /.5.

100 Chae : The Thi i Fig. 3-. ongiudinal Couling medane, imaginay a: βγ., /.4, /.5. Fig. 3-. ongiudinal Couling medane, eal a: βγ., /.6, /.5. 3

101 Chae : The Thi i Fig ongiudinal Couling medane, imaginay a: βγ., /.6, /.5. Fig ongiudinal Couling medane, eal a: βγ, /., /.5. 4

102 Chae : The Thi i Fig ongiudinal Couling medane, imaginay a: βγ, /., /.5. Fig ongiudinal Couling medane, eal a: βγ, /.4, /.5. 5

103 Chae : The Thi i Fig ongiudinal Couling medane, imaginay a: βγ, /.4, /.5. Fig ongiudinal Couling medane, eal a: βγ, /.6, /.5. 6