4.1 The Transverse Resonance Diffraction (TRD) approach

Transcription

1 8 ispese 4. The Tsvese Resoce Dicio TRD ppoch Tsvese Resoce i is bsic om s is iouce b Coh [4] i oe o lse slolie sucue pl veguie popose b himsel. We hve le me is eel piciple i Chpe he e use i o he chceisio o ielecicll loe ecgul veguie. The piciple s evelope i oe o ccou o iscoiui o he geil mgeic iel some ple mel cosieig some iiiel hi mellisio. I sevel s he Tsvese Resoce o pl veguies is lie o he Specl Domi Appoch SDA o be iouce i he olloig secio. As he SDA i elies o he hpohesis o homogeei i oe iecio so h iel iscoiuiies m ol be iouce i ples pllel o he iecio o homogeei. Cosequel ol iiiel hi mellisios m be cosiee. Le i s geelise io Tsvese Resoce Dicio hich hs bee use o ie ge o sucues such s o emple he ise ielecic veguie [5] hee he geomeies bove belo he iscoiui e o ecessil he sme. I hs bee use i ve ie ge o sucues. I oe o epli is bsic piciples le us ppl i o iiiel hi lossless micosip Figue 4.. b egio egio Figue 4. Iiiel hi micosip lie I Secio. e ou h mgeic elecic iels o he boe ielecic sc e lie b veco Gee ucio i ech homogeeous egio h is o egio H hile o egio H Y E 4. Y E 4. I he bove e hve eplicil iice he epeece o he Gee ucios o he logiuil popgio cos o he equec b mes o o.

2 Iel pl veguies o mulilee subse: D lsis 9 The eils o compuio o he Gee ucios hve bee evelope i Chpe bu he e o iecl o iees o ou pese iscussio s e e essig he poblem moe geel level. Wh is impo e ssume h hese Gee ucios li elecic mgeic iels guee h bou coiios o he eclosig bo e sisie. Hece e hve sill o impose he bou coiios o he iece beee mei oe o. To his e e subc he iece obiig H H [ Y Y E E ] The svese elecic iel he iece mus be coiuous so h 4.3 eq. 4.3 m be eie s H H E E E 4.4 [ Y Y ] E 4.5 The bove elioship m be ivee i oe o elici he svese elecic iel s ucio o he ieece beee he svese mgeic iels: he meig o such ivesio s ee he e o secio o. Oe obis hee [ H H ] E 4.6 [ Y Y ] o he svese mgeic iels oo e coiuous evehee bu o he mel sip hee suce elecic cue shee is iuce b he mgeic iel iscoiui h is [ H H ] [ H H ] 4.7 beig he ui veco oml o he mel suce. Cosequel he svese iel secio m be lie o suce cue o-vishig jus o he sip suce: Sip E 4.8

3 ispese I is sigho o vei h opeo i 4.8 is lie o b mes o he elioship: 4.9 Thee is sill bou coiio le ou o ou iscussio: he elecic iel geil o he sip couco mus vish h is so h 4.8 becomes E i Sip 4. i Sip 4. Sip h is homogeeous iegl veco equio o be solve i he sip egio. Is soluio iels he popgio coss o ech moe suppoe b he sucue s ell s hei ispesio chceisics he suce couco cues o les hei spil isibuios. Hoeve solvig iegl equio m o be simple s: ue o is impoce e ill eice o he subjec pgph he e oe. A e moe cosieios bou 4.: is oe h i is so o implici eigevlue equio hose eigevlues e he uo popgio coss o give equec hees he spil cue isibuios e he eigeucios. Is implici ue s e ill see m be seious bc. Moeove oe h i oul be omll coec o sop ou mhemicl mipulio o equio 4.5 so obiig b viue o 4.7 Y E 4. The geil elecic iel is o-vishig jus ousie he sip egio hee he suce elecic cue is eo so e obi he eigevlue iegl equio Gp Y E 4.3 o be solve ousie he sip egio mel he gp egio. This is ul omulio o 4.: he bove seme becomes ppe heeve e ppl Bbie s piciple o compleme scees see e.g. [6]. I c e c iouce mgeicll coucig scee o he gp egio suppoig mgeic souce M E

4 Iel pl veguies o mulilee subse: D lsis 4.3 m be eie i ems o mgeic cues solve o he mgeic couco. I is ve es o chec he peec equivlece o he o coiios h is suce elecic cue o elecic sip o suce mgeic cue o mgeic sips eeig ove he compleme suce o he elecic sip. I c eleme o elecic cue pouces o compoe o mgeic iel geil o ple coiig he eleme e.g. ipole e so h he mgeic geil iel o he gp egio mus vish his is esue b he ioucio o he mgeic scee. O he ohe h eleme o mgeic cue pouces o compoe o elecic iel geil o ple coiig he eleme so h he ssume mgeic cue ovishig jus o he gp ill iisicll esue he vishig o he geil elecic iel o he sip suce. Hece e c choose iieel o solve eihe 4. o 4.3. A les heoeicll. Acull i us ou h solvig 4. is moe coveie o micosip: his coclusio epes o he iegl equios e solve i pcice. As quie geel ule oe hs o choose he equio hose uo qui is eie ove he smlle omi: so o micosip he suce cue m be speciie moe esil h he geil elecic iel ove he compleme egio. Vice ves mgeic souce omulio is coveie o copl veguies o slo-lies. Fo oe c see ho o copl veguie oe ees o cosie he suce elecic cue ove hee egios o o hich e iiie he sip coucos o suce mgeic cue ove jus o iie egios he ieelecoe gps. 4. O he soluio o iegl equios This is sho mhemicl pehesis o help ues some o he poblems lie ih he soluio o he iegl equios h e occuig. Thee e umbe o mhemicl boos essig he poblem om geel poi o vie: complee eme is pesee b D.S. oes [7]. A ieesig ppe b S [8] povies h summ useul iscussio bou he mehos mos iel use i elecomgeics. Oe elecomgeic poblems m be epesse i ems o he opeo equio A 4.4 hee A is iego-ieeil opeo. The mos popul ppoches o his poblem e he so clle Rleigh-Ri viiol meho he meho o momes [9] he Glei meho he meho o les sques. All hese echiques pocee b ssumig pioi h he uo qui m be

5 ispese epesse s supeposiio o o epig ucios: he ie is h ppoime soluio o m be epesse s u 4.5 hee he e uo coeicies o be solve o. The epig ucios u e complee bsis i -imesiol spce: oe h i is o ecess h he om ohogol se o ucios. The ec soluio lies i iiie imesiol spce e ill cosie he ppoime soluios o covege o he el soluio i lim 4.6 This is sog covegece oe clle covegece i he om o covegece i he me is meig is h o give ie > e c i umbe such h < 4.7 I he viiol Rleigh-Ri meho he soluio is obie b miimisig he uciol F eie b F A 4.8 hee he bces e use o iice he ie pouc. I m be pplie i his om ol i A is sel-joi opeo h is i sisies he equli I c i his cse 4.8 m be eie s A A 4.9 F A 4. i ol i A is posiive eiie opeo h is i sisies he coiio miimisio o F iecl implies he miimisio o he em be Au u > u 4.. I c i 4. oes o epe o hile ue o he eiieess o A i mus c A 4. hee c is peeemie posiive cos. Hece i he hpoheses e sisie e pocee b eplcig b he ppoime o 4.5 i he eiiio 4.8:

6 Iel pl veguies o mulilee subse: D lsis 3 F Au u u u 4.3 i The uciol F is miimise b equig he pil eivives i i F i i K o eo. This e ecove se o lgebic equios i Au u u K 4.4 hose soluio iels he uo coeicies. oe h hee is o obvious ee o he epig ucios o sis he coec bou coiios: he ee o be i he omi o he opeo. Hoeve hee e umbe o iiol cosis i oe o ou iscussio o ppl: he opeo A mus be o-sigul is ivese mus be bou he epig ucios mus om complee se ih espec o he ie pouc Au i u j Moeove s e hve see he opeo mus be posiive eiie o hee is o guee h b eploiig lge se o epig ucios he ppoime soluio mus be close o he ec oe i he sese o he me covegece. The impo poi is h ue he bove speciicios he il soluio oes o ee o sis he bou coiios. oeheless i his cse miimisio o F ill o guee miimisio o he coespoig esiul [A ]: e co eve se h hese esiuls ill ecese o eo s iceses o epsio se o belogig o he omi o he opeo. Thee is moe iec o solve equio 4.4: e ise ou il epsio 4.5 iecl i 4.4 e loo o he uo coeicies hich m solve Au 4.5 To his im e iouce uhe se o elemes... M i he ge o A peom he ie pouc o 4.5 b ech o he m so h Au m... M 4.6 m he coeicies e obie om he soluio o he lgebic m ssem. This meho is o b vious mes: i oe is eee o s meho o momes picull he smmeic ie pouc is ivolve i.e. o m i i

7 4 ispese comple cojuge he seco em isig om he elecomgeic cocep o ecio. I his cse he sme equios m be eive om ecipoci heoems. oe h he s scl pouc h is he oe ivolvig he comple cojuge is lie o he cocep o poe he viiol meho o h o miimisig he ssem eeg. o he s scl pouc ull povies eiiio o om llos ig m o he cosequeces e hve se so bou covegece o ppoime soluio o he ue soluio. oeheless s e hve see i Chpe he eigeucios o loss veguie m be ohogol jus ih espec o he scl pouc eive om ecipoci hile o beig ih espec o he s scl pouc. This c highlighs h he smmeic ie pouc is so use i elecomgeics. I he mhemicl lieue 4.6 is o s geel Glei meho o Peov-Glei meho. Geell hee is some geeme i simpl eiig Glei meho s he picul cse o M u. B compig he equios o be solve i he picul cse o he Glei meho ih he oes obie b he viiol meho 4.4 i ppes h he e omll ieicl. oeheless hee e some umel ieeces: he opeo A oes o ee o be sel-joi; i c eve be o-lie! O couse i such cse 4.4 oul jus be eplce b A u u u K 4.7 Bu ohig comes o ee: he epig ucios mus be bsis o boh he omi he ge o A. All ucios u mus sis he bou he ieeibili coiios o he poblem. I he bou coiios e o sisie he iiol cosis o he soluio poceue hve o be impose. Geell he Glei meho jus eoces e covegece o he esiuls h is i esues h A u o ll u. O he ohe h i he opeo is boue i A C ih C cos iepee om he sog covegece o he esiuls is guee h is lim A I he ls cse i he ivese opeo is lso boue he he sequece o he ppoime soluios oo coveges sogl o he ec soluio.

8 Iel pl veguies o mulilee subse: D lsis 5 As geel cosieio povie b [8] he Glei meho is suible o poblems ivolvig iegl opeos s hese e oe boue equiemes o he ieeibili o he epig ucios e ee h i poblems ivolvig ieeil opeos. Someimes he Glei meho is lso use i eigevlue poblems o be ee e...; hoeve s eigevlue poblems e o-lie eve he he opeo is lie he heo becomes moe moe elboe i lls ousie ou pese coce. The mi esul is h se o coiios is o esuig he covegece o ppoime eigevlues evlue b he Glei meho o he cul eigevlues o he opeo. The ieese ee m i iiol iomio i he eeeces. Fo he se o compleeess e hve o el lile ih oe iiol meho: he meho o les sques. The ie is o emp o miimise he uciol G A A A h is o impose esiuls o be miimum i he me sques sese. The esulig se o equios is he Au Au Au I is cle h seig up he les sques meho is hevie h ppochig he poblem b he Glei meho. Is vge is h esiuls o covege moooicll i he me sques sese o eo jus povie h oe selecs he u so h Au om liel iepee complee se. Fill i he u e eigeucios o A h his meho omll euces o he Glei oe compe 4.8 o 4.4 ue his hpohesis. 4.3 Soluio o he TRD iegl equio Le us cosie iiiel hi shiele micosip lie uiom i he popgio iecio: e hve see h he mos suible iegl equio o his cse povie b he TRD ppoch is 4.. The le m be solve b mes o he Glei meho. To his e le us ep ech uo cue compoe b se o suible ucios h is 4.9

9 6 ispese A possible se o epig ucios sisig he bou coiios lso ccouig o he sigul behviou o he mgeic iel compoe oml o he siplie ege is si [ ] cos [ ] 4.3 The siguli o he mgeic iel oml o he siplie ege is cosequece o o ssumpios mel sip iiie coucivi eo ius o cuvue Figue 4. Shpes o he epig ucios We ill el ih hese siguliies i moe geel coe i he e chpe. A em bou epsio 4.3: he ucios ivolve i 4.3 e ele o he Chebshev polomils iel use i sip slo poblems s he usull o ve ell: he eso o such success is h Chebshev

10 Iel pl veguies o mulilee subse: D lsis 7 polomils e sic ppoimios o Mhieu ucios hich i us e igoous soluios o he ve equio i he ellipicl co-oie ssem i hich he sip c be escibe ecl s equico-oie suce [see e.g. ]. The use o 4.3 i plce o he cul Chebshev polomils is jusiie b hei simple Fouie som. I eiig he epig se e hve slighl chge he eeece ssem so h is he cee o he sip: his mes he compuio o he iegls equie b he e sep o he Glei poceue bi esie. The behviou o 4.3 is epice i Figue 4.. The epig se is he subsiue i he iegl equio o be solve h is 4. obiig: 4.3 Fo he se o compleeess e epo he epessios o he opeo obie b eploiig b some mipulios: [ ] j LSM LSE h h LSE LSM h e LSM LSE e e φ φ φ φ φ φ 4.3 I he bove e hve eie co co b j j 4.33

11 8 ispese ccoig o.7.7 LSM i LSE i ω i i i i ωµ Fill le us ecll h ue o.64 he -popgio coss e i i h φ e φh e eie b.75; hece e hve ll e ee i oe o solve he poblem. Iciell i oe compes he bove epessios ih he oes povie b.4 i is ppe h some i o sog elioship mus eis: i c he ic opeo o.4 seems o be geel om o 4.3 *. We jus oe e ieeces: he em liig he he compoes o.4 coi em o ppeig hee. This is jus he cosequece o he eeece ssem compe Figue 4. Figue.3 he coseque iee choice o he LSM LSE poeils Iee i e o use he eeece ssem hee is he bo cee ccoig o he epig ucios e hve jus eie e o ee he em he poeils eie i.95. Moeove he geel omule coi γ ise o essig he geel loss cse o hich γ α j The scl Gee ucios ppeig i.4 euce igh o o epessio 4.33 he boh he souce he es poi mel e o he sme ple s o couse i his simple cse o jus o ielecic les!. Fill oe h 4.33 s obie b imposig ll he bou coiios log he -iecio o he poeils o.6 ihou eeece o he svese lie impece cocep. This bes iess o he impoce o.4 ele heo o he scl Gee ucios. Le us come bc o ou oigil poblem: he ls sep o he Glei poceue is o compue he ie pouc o 4.3 ih ech o he epig ucios. mel e peom he ie pouc o he is o 4.3 b beig.. he seco b h is * Acull i iig.4 e hve use slighl iee oio o he scl poeils: i mus be cosiee h Ψ sems o φh Φ o φ e.

12 Iel pl veguies o mulilee subse: D lsis This e obi lgebic ssem o be solve o he uo coeicies. The iegls o be peome e sigho s he o vibles e o couple: o i 4 eve i e φ 4 eve i h φ 4.35 I he bove epessios eoes he eo-oe Bessel ucio o he is i. We pu he lgebic ssem i mi om obiig 4.36 beig [ ] T Hece ll is much simple h i m loo is glce. All e ee o is o-ivil soluio o 4.36 so h e loo o he soluios o e 4.37 hich e he eigevlues h is he moe popgio coss o he sucue. Fo ech eigevlue e ge he coespoig eigeveco seig o emple

13 3 ispese obiig o give he emiig compoe o he eigeveco om he o eemiisic ssem. This he ppoime cue isibuio is obie he mol iels m be clcule evehee i he bo. Le us e moe eile loo eq. 4.37: he pim souce o is eos is h p o he mi eive om he opeo. I picul he eigevlues e cosiue b hose vlues o lloig he LSM impece see eq. 4.3 o blce he LSE impece. This c is ppe o he smpoic p o he sum h is o he ems compue o high vlues o he ie : s e hve eme LSE LSM sepio iece b ech LSELSM moe. As becomes lge i is sigho o sho h so h LSM LSE LSM i LSE i e he impeces see he j i ω j j i ωµ j ω ω µ j µ µ µ Hece i is ppe om 4.3 h i he opeo he LSM he LSE ps emi ls compble mig i possible o he LSM p epeig o o ccel he LSE p. O he ohe h o emple i he opeo he LSE p becomes pil egligible ih espec o he LSM p he possibili h he seies sums o vish o some eigevlue us ou o be uliel. The sme pplies o he emiig opeos. This ough gumeio jus highlighs h he mi cue compoe is. Cosequel le us ppl s eecise he TRD meho o he iel

14 Iel pl veguies o mulilee subse: D lsis 3 micosip cosieig jus oe epig ucio. Ye le us ssume h he sip is o ih espec o he velegh so h Hece he eigevlues ill simpl be he soluios o he scl equio 3.. LSM LSE 4.4 I is ell sigho o se up compue pogm o solve 4.4 o o solve i b mes o oe o he cuel vilble geel-pupose mhemics pogms. Resuls obie b 4.4 m be compe ih oe o he ppoime close omule vilble i he lieue: ccoig o he esuls b Wheele [] epoe i []:.4 i i F F e 4.4 These esuls o o e io ccou he equec ispesio o he chceisics. To his e he moiicio suggese b Kobshi [3] is: m e e hee > ep ].33 [ i i m m m m m c c c e e e TM TM