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1 ARISTOTELEIO PANEPISTHMIO JESSALONIKHS POLUTEQNIKH SQOLH TMHMA HLEKTROLOGWN MHQANIKWN & MHQANIKWN UPOLOGISTWN TOMEAS THLEPIKOINWNIWN Diplwmatik ErgasÐa tou Papadìpoulou N. Iw nnh Melèth thc 'AllhlepÐdrashc EpÐpedwn Kum twn kai Diamorfwmènwn Epifanei n Tèleiou Met llou me th Mèjodo Prosarmog c Rujm n Epiblèpwn Kajhght c: Emmanou l Kriez c ΘΕΣΣΑΛΟΝΙΚΗ Μάρτιος 29

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3 Prìlogoc Xekin ntac th diadikasða thc diplwmatik c ergasðac, perðpou èna qrìno prin, h e- reunhtik perioq thc plasmonik c me exèplhxe me ton meg lo arijmì twn ereunhtik n rjrwn pou thn aforoôsan. 'Eqw antilhfjeð plèon, èna qrìno met, ìti autì, to exairetik meg lo endiafèron, phg zei apì thn plhj ra kai th diaforetikìthta twn pijan n efarmog n pou mporoôn na prokôyoun apì ta apotelèsma autoô tou endiafèrontoc. H paroôsa ergasða apoteleð ousiastik, thn eisagwg mou ston q ro twn upologistik n teqnik n tou hlektromagnhtismoô. Xekin ntac apì basik stoiqeða thc jewrðac thc plasmonik c to jèma thc ergasða, topojethmèno ìso pio genik gðnetai, eðnai h diaqeðrish thc allhlepðdrashc epðpedwn hlektromagnhtik n kum twn kai diamorfwmènwn epifanei n tèleiwn met llwn, me th qr sh thc mejìdou prosarmog c rujm n kai hmelèthtwn fainomènwn pou prokôptoun. EÐnai shmantikì gia mèna, na euqarist sw, ton epiblèponta thc diplwmatik c mou ergasðac, kajhght tou tm matoc, k. Emmanou l Kriez, tìso gia th sunolik tou prosfor sthn akadhma k mou poreða, ìso kai gia thn amèristh bo jeia kai kajod ghsh apì thn pr th stigm ekpìnhshc thc paroôsac ergasðac. EpÐshc, jèlw na euqarist sw ton kajhght tou tm matoc, k. Tra anìgioôltsh, gia th bo jeia tou apì thn pr th stigm thc eisagwg c mou ston tomèa Thlepikoinwni n tou tm matoc kai tèloc ton upoy fio did ktora tou tm matoc, k. Odussèa Tsilip ko, gia thn polôtimh bo jeia tou sto telikì st dio thc suggraf c. JessalonÐkh, M rtioc 28 Iw nnhc N. Papadìpouloc i

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5 Perieqìmena Prìlogoc Perieqìmena i iii 1 Eisagwg Genik perð PolaritonÐwn Epifaneiak n PlasmonÐwn kai efarmogèc PerÐlhyh thc ergasðac Uposthrizìmenoi RujmoÐ sth Diepif neia Met llou DihlektrikoÔ SÔntomh enaskìphsh twn hlektromagnhtik n idiot twn twn met llwn Majhmatik perigraf twn rujm n ProsomoÐwsh EpibebaÐwsh twn SPP rujm n Di gramma diaspor c SPP rujm n Diègersh SPP rujm n sth diepif neia met llou dihlektrikoô SÔzeuxh mèsw prðsmatoc SÔzeuxh me th qr sh fr gmatoc par jlashc anaglôfou Metaôlik arnhtikoô deðkth di jlashc Genik perð metaôlik n arnhtikoô deðkth di jlashc Mèjodoc prosdiorismoô energoô deðkth di jlashc n eff ProsomoÐwsh di taxhc arnhtikoô deðkth di jlashc Mèjodoc Prosarmog c Rujm n Genikèc idiìthtec epðpedwn kum twn An lush epðpedou kômatoc se s kai p pìlwsh An lush rujm n Floquet Bloch Exagwg sunistws n magnhtikoô pedðou Efarmog thc mejìdou prosarmog c rujm n Efarmog thc mejìdou sthn perðptwsh mon c periodikìthtac Metallik epif neia me orjogwnik aul kia TM rujmoð Metallik epif neia me orjogwnikèc egkopèc TM rujmoð Efarmog thc mejìdou sthn perðptwsh dipl c periodikìthtac Analutikèc ekfr seic pedðwn iii

6 5 Yeudo SPP rujmoð kai exairetik optik met dosh Apì tic optikèc suqnìthtec stic mikrokumatikèc, apì touc SPP stouc yeudo SPP rujmoôc Exairetik Optik Met dosh Metallikì fôllo me orjogwnik anoðgmata TM pìlwsh Metallik epif neia me orjogwnikèc opèc Suzeugmènoi yeudo SPP rujmoð Yeudo SPP rujmoð sth diepif neia met llou me tetragwnikèc opèc peirou b jouc kai dihlektrikoô Yeudo SPP rujmoð sth dipl diepif neia tèleiou met llou me tetragwnikèc opèc kai dihlektrikoô Sumper smata BibliografÐa 84 iv

7 Kef laio 1 Eisagwg 1.1 Genik perð PolaritonÐwn Epifaneiak n PlasmonÐwn kai efarmogèc Hepisthmonik perioq thc plasmonik c, mia enìthta tou genikìterou tomèa thc nanofwtonik c, èqei apokt sei polô meg lh dunamik ta teleutaða qrìnia, sugkentr nontac meg lo arijmì dhmosieôsewn. To meg lo autì ereunhtikì endiafèron aitiologeðtai apì thn exairetik shmasða pou èqoun oi pijanèc efarmogèc thc teqnologðac. O pur nac thc ereunhtik c perioq c thc plasmonik c apoteleðtai ousiastik apì ta polaritìnia epifaneiak n plasmonðwn (surface plasmon polaritons, SPP). Ta surface plasmon polaritons eðnai hlektromagnhtikèc diegèrseic pou diadðdontai sth diepif neia metaxô agwgoô kai dihlektrikoô. AutoÐ oi epifaneiakoð hlektromagnhtikoð rujmoð prokôptoun mèsw thc sôzeuxhc hlektromagnhtik n kum twn me suntonizìmena hlektrìnia thc stoib dac agwgimìthtac tou met llou. Me b sh aut ta epifaneiak kômata prokôptoun mia seir polô shmantik n efarmog n, merikèc apì tic opoðec ja analujoôn sth paroôsa ergasða. Ta basik stoiqeða pou telik sunjètoun thn ereunhtik aut perioq tan dh gnwst apì tic arqèc tou 2 ou ai na. Oi ergasðec tou Paul Drude, to 19, èjesan tic b seic ga thn perigraf thc hlektromagnhtik c sumperifor c twn met llwn stic optikèc suqnìthtec mèswtou gnwstoô montèlou Drude. Thn Ðdia qronik perðodo jemeli jhke h majhmatik perigraf thc sumperifor twn epifaneiak n aut n kum twn pou emfanðzontai sth diepif neia metaxô met llou peperasmènhc agwgimìthtac kai dihlektrikoô stic ergasðec twn Sommerfeld to 1899 kai Zenneck to 197. EpÐshc, to 192 mèsw twn ergasi n tou Wood, parathr jhke to fainìmeno thc mh anamenìmenhc pt shc tou suntelest an klashc fwtìc, sthn perioq tou optikoô f smatoc, kat thn an klash tou se metallik epif neia periodik diamorfwmènh, to gnwstì metallikì fr gma par jlashc (metallic grating), qwrðc ìmwc na sundejeð me tic prohgoômenec jewrhtikèc ergasðec perigraf c twn epifaneiak n kum twn. H sôndesh aut ègine sqedìn misì ai na argìtera stic ergasðec tou Ritchie to 1957, opìte kai jemeli jhke gia pr th for h Ôparxh twn rujm n SPP. H Ðdia h fôsh twn rujm n aut n, suzeugmènwn sthn epif neia met llou dihlektrikoô, dðnei th dunatìthta kumatod ghshc tou fwtìc se tètoiec diat xeic. Me dedomèno ìti h perioq sugkèntrwshc tou rujmoô eðnai mikrìterh tou m kouc kômatoc, 1

8 Kef laio 1. Eisagwg 1μm (αʹ) Γεωμετρία μεταλλικής ταινίας θαμμένης σε διηλεκτρικό. (βʹ) Εικόνα μικροσκοπίου της έντασης του φωτός του βασικού LR SPP ρυθμού στην έξοδο της διάταξης. Sq ma 1.1: GewmetrÐa di doshc kai eikìna tou rujmoô sth diatom. ìpwc prokôptei apì tic sqetikèc analôseic, antilambanìmaste ìti to Ðdio to fainìmeno epitrèpei th dhmiourgða exarthm twn kai diat xewn pou xepernoôn to ìrio par jlashc (diffraction limit). Gia poll qrìnia to pleonèkthma twn hlektronik n kuklwm twn ènanti twn fwtonik n tan to polô mikrì touc mègejoc (tupikì mègejoc transistor < 1nm, mègejoc fwtonikoô kukl matoc merik mikrìmetra). Apì thn llh pleur, ta fwtonik kukl mata ekmetalleôontai pl rwc thn taqôthta di doshc tou fwtìc, pou eðnai pollèc t xhc megalôterh thc taqôthtac di doshc tou hlektrikoô reômatoc sta hlektronik kukl mata, epitrèpontac ètsi polô megalôterec taqôthtec met doshc twn shm twn. H kumatod ghsh tou fwtìc me th qr sh twn SPP èrqetai na kalôyei to q sma autì epitrèpontac th di dosh tou fwtìc se klðmakec mikrìterec tou m kouc kômatoc. Epiplèon, h qr sh twn metallik n tmhm twn gia th kumatod ghsh tou fwtìc epitrèpei thn tautìqronh qr sh tou Ðdiou diaôlou} gia thn di dosh tou optikoô s matoc plhroforðac kai th dièleush hlektrikoô s matoc elègqou. H dièleush tou hlektrikoô s matoc elègqou mèsw tou Ðdiou di ulou epitrèpei thn kalôterh apotelesmatikìthta tou elègqou (p.q. aôxhsh apodotikìthtac jermoptikoô fainomènou), kaj c to s ma elègqou brðsketai ston pur na thc di doshc kai ìqi se k poia perioq gôrw apì aut n. To meionèkthma ìmwc thc kumatod ghshc me th qr sh twn SPP prokôptei apì tic uyhlèc ap leiec pou parousi zoun oi rujmoð, lìgw thc uyhl c dieðsdushc sto eswterikì twn metallik n tmhm twn. Gia thn kumatod ghsh tou fwtìc sthn epif neia twn met llwn èqoun protajeð di - forec gewmetrðec me diaforetik qarakthristik. Sto sq ma 1.1(a ) parousi zetai h gewmetrða metallik c tainðac jammènhc} se dihlektrikì, en sto sq ma 1.1(b ) h eikìna optikoô mikroskopðou thc èntashc tou fwtìc sthn metallik tainða. Sta plaðsia aut c thc gewmetrðac, oi dôo rujmoð pou dhmiourgoôntai stic dôo epif neiec epaf c thc meg lhc pleur c thc metallik c tainðac me to uperkeðmeno dihlektrikì, mporoôn na suzeuqjoôn metaxô touc dhmiourg ntac èna rujmì pou ekteðnetai kurðwc sto dihlektrikì kai mporeð na diadojeð se axiìlogo m koc. O rujmìc autìc apokaleðtai Long Range SPP (LR SPP). Mia llh prosèggish eðnai aut pou jumðzei th gewmetrða mikrotainðac RF suqnot twn qwrðc ìmwc thn parousða tou ag gimou fôllou sth b sh. H gewmetrða 2

9 Kef laio 1. Eisagwg y z W x μm, διαστάσεις ανοίγματος : 5~1nm (αʹ) Διάταξημεταλλικής ταινίας πάνω σε διηλεκτρικό. (βʹ) Διάταξημεταλλικού φύλλου με εγκοπή: ησυγνέντρωσητου ρυθμού γίνεται εντός του ανοίγματος. W=45nm (γʹ) Εντασηπεδίου στηδιατομή για την περίπτωση της γεωμετρίας του σχήματος 1.2(αʹ) (δʹ) Προσομοίωση της έντασης το πεδίου στηδιάταξητου σχήματος 1.2(βʹ). Sq ma 1.2: Diadedomènec gewmetrðec kumatod ghshc SPP rujm n kai katanom thc èntashc tou pedðou. aut parousi zetai sto sq ma 1.2(a ). Shmantik prosfor sth jewrhtik an lush twn rujm n pou uposthrðzontai apì tic parap nw diat xeic eðqan oi ergasðec tou Pier Berini [1, 2]. Tèloc, mia diaforetik gewmetrða eðnai h sumplhrwmatik twn prohgoômenwn ìpou o rujmìc sugkentr netai se mða egkop metaxô dôo metallik n fôllwn, ìpwc parousi zetai sto sq ma1.2(b ). Oi rujmoð SPP mporoôn na qrhsimopoihjoôn gia th beltðwsh thc euaisjhsðac twn optik n mikroskopðwn. Sta plaðsia thc optik c mikroskopðac to ìrio par jlashc eis gei èna jemeli dh periorismì sthn an lush thc lambanìmenhc eikìnac. Kat thn an klash tou fwtìc apì to epithroômeno antikeðmeno èna mèroc thc plhroforðac eðnai suzeugmèno se kômata me meg lo kumatikì arijmì kai ra kômata pou telik aposbènontai kaj c diadðdontai sto dihlektrikì, me apotèlesma thn ap leia mèrouc thc plhroforðac. Me th qr sh twn SPP, èqoume th dunatìthta na xeper soume kat èna trìpo autìn ton periorismì, kaj c ta aposbenômmena kômata me meg lo kumatikì arijmì mporoôn na suzeuqjoôn me SPP rujmoôc se kat llhlh diepif neia kai sth sunèqeia h plhroforða mporeð na exaqjeð apì touc rujmoôc autoôc. H allhlepðdrash tou fwtìc me ta mètalla mporeð na odhg sei se nèa ulik me e- xairetikèc idiìthtec. 'Etsi, gia par deigma, periodikèc diat xeic metallik n stoiqeðwn kai dihlektrikoô mporoôn na dhmiourg soun nèa ulik pou emfanðzoun arnhtikì deðkth 3

10 Kef laio 1. Eisagwg di jlashc stic optikèc kai llec suqnìthtec. Ta ulik aut, gnwst wc metaôlik arnhtikoô deðkth di jlashc (negative index metamaterials), gennoôn mða nèa ereunhtik perioq idiaðterou endiafèrontoc. Tèloc, h Ðdia allhlepðdrash tou fwtìc me ta mètalla genn to fainìmeno pou parathr jhke apì ton Ebbessen [3] kai onom sthke exairetik optik met dosh (extraordinary optical transmission). To fainìmeno perigr fei thn parat rhsh suntelest n met doshc pou xepernoôn kat polô tic jewrhtikèc problèyeic gia diat xeic di trhtwn metallik n fôllwn stic opoðec prospðptei optikì kôma. Ja perigr youme k poiec apì tic idiìthtec kai tic efarmogèc pou prokôptoun apì touc rujmoôc SPP kai ja prospaj soume na exhg soume to lìgo gia ton opoðo h meg lh ereunhtik prosp jeia sthn perioq thc plasmonik c den apoteleð apl mða mìda twn hmer n, all mporeð na prosfèrei nèec diexìdouc se eggeneðc periorismoôc twn rimwn teqnologi n thc fwtonik c. 1.2 PerÐlhyh thc ergasðac Sta plaðsia aut c thc ergasðac ja asqolhjoôme kurðwc me thn allhlepðdrash twn hlektromagnhtik n kum twn me metallikèc epif neiec kai tic idiìthtec pou prokôptoun apì aut thn allhlepðdrash. DÔo polô shmantik zht mata pou prokôptoun eðnai, ìpwc anafèrame kai parap nw, ta metaôlik arnhtikoô deðkth di jlashc kai to fainìmeno thc exairetik c optik c met doshc. Sto parìn kef laio, kef laio 1, gðnetai mia polô sôntomh istorik anadrom sth jewrða twn rujm n SPP kai mia mia sôntomh anafor twn idiìthtwn kai pijan n efarmog n touc. Sto kef laio 2, ja perigrafoôn analutik oi uposthrizìmenoi SPP rujmoð sth mon diepif neia met llou dihlektrikoô. H morf twn rujm n ja prokôyei apì thn analutik epðlush twn exis sewn tou Maxwell gia th sugkekrimènh gewmetrða. Sth sunèqeia ja parousiastoôn k poiec sqetikèc prosomoi seic me th qr sh thc mejìdou peperasmènwn stoiqeðwn prokeimènou na epibebai soume thn analutik morf twn rujm n pou prokôptoun. Sto tèloc tou kefalaðou perigr fetai h sqèsh diaspor c twn rujm n aut n, kaj c kai oi trìpoi kai oi diat xeic diègershc twn rujm n aut n sth diepif neia met llou dihlektrikoô. Sto kef laio 3 ja perigr youme tic diat xeic twn metaôlik n arnhtikoô deðkth di jlashc. Pèra apì thn eisagwg sth basik jewrða twn metaôlik n aut n, ja perigrafeð h mèjodoc twn Soukoulis, Markos, Smith [4] gia thn exagwg tou energoô deðkth di jlashc apì touc suntelestèc an klashc kai met doshc twn diat xewn aut n. Tèloc, to kef laio ja oloklhrwjeð me thn prosomoðwsh kai th efarmog thc mejìdou aut c se periodik di taxh metallik n fôllwn se dihlektrikì, h opoða ìpwc ja apodeiqjeð parousi zei arnhtikì pragmatikì mèroc tou deðkth di jlashc stic optikèc suqnìthtec. Sto kef laio 4, ja perigrafeð h mèjodoc prosarmog c rujm n (Mode Matching Method). H mèjodoc aut ja epitrèyei sth sunèqeia na analôsoume tic diat xeic allhlepðdrashc hlektromagnhtik n kum twn me diamorfwmènec metallikèc epif neiec. Ja efarmosteð h mèjodoc se peript seic met llwn me monodi stath periodikìthta, orjogwnik anoðgmata kai aul kia, kai me disdi stath periodikìthta ìpwc metallikì 4

11 Kef laio 1. Eisagwg fôllo me orjogwnikèc opèc. Tèloc, sto kef laio 5 ja prosomoiwjoôn diat xeic allhlepðdrashc diamorfwmènwn metallik n fôllwn me epðpeda hlektromagnhtik kômata me th qr sh thc mejìdou prosarmog c rujm n, h opoða analôjhke sto kef laio 4. Sta apotelèsmata twn prosomoi sewn perilamb nontai h parat rhsh tou fainomènou thc exairetik c optik c met doshc kai oi yeudo SPP rujmoð, dhlad rujmoð pou omoi zoun me touc SPP rujmoôc, pou emfanðzontai sthn epif neia diamorfwmènou tèleiou agwgoô. 5

12 Kef laio 2 Uposthrizìmenoi RujmoÐ sth Diepif neia Met llou DihlektrikoÔ Sto parìn kef laio, me b sh thn jewrða tou Maxwell, ja proqwr soume sthn exagwg thc analutik c èkfrashc twn rujm n pou uposthrðzontai apì th mon diepif neia met llou peperasmènhc agwgimìthtac kai dihlektrikoô kaj c kai thc antðstoiqhc sqèshc di sporac twn uposthrizìmenwn rujm n. Sth sunèqeia ja sugkrðnoume ta analutik apotelèsmata, me ta antðstoiqa pou prokôptoun apì thn epðlush thc diepif neiac me b sh th mèjodo peperasmènwn stoiqeðwn kai tèloc ja parousi soume tic mejìdouc kai tic diat xeic diègershc twn rujm n aut n. To kef laio ousiastik apoteleð mða eisagwg ston aplì qeirismì k poiwn gewmetri n me qr sh mejìdwn tou upologistikoô hlektromagnhtismoô kai th sôgkrish twn exag menwn apotelesm twn. 2.1 SÔntomh enaskìphsh twn hlektromagnhtik n idiot twn twn met llwn H melèth twn hlektromagnhtik n idiot twn twn met llwn apoteleð kat kôrio lìgo antikeðmeno thc epist mhc thc fusik c stere c kat stashc kai ra protrèpetai o anagn sthc na apeujunjeð se antðstoiqa suggr mmata prokeimènou na brei mia oloklhrwmènh prosèggish tou jèmatoc. Sta pl aðsia aut c thc ergasðac ja anaferjoôme se k poiec basikèc arqèc pou perigr foun thn hlektromagnhtik sumperifor twn met llwn, kaj c stic idiìthtec autèc sthrðzetai h Ðdia h jewrða twn plasmonik n rujm n kai ta fainìmena pou sqetðzontai me autoôc, antikeðmena pou ja mac apasqol soun sth sunèqeia. Shmantikìterh sumbol sthn ex ghsh twn hlektromagnhtik n idiot twn twn met llwn stic uyhlèc suqnìthtec eðqe h ergasða tou Paul Drude to SÔmfwna me to montèlo pou prot jhke, gnwstì kai wc montèlo Drude, h sumperifor enìc met llou 1 Η σχετική εργασία έχει τίτλο Zur Elektronentheorie der Metalle; II. Teil. Galvanomagnetische und thermomagnetische Effecte και δημοσιεύτηκε στο Annalen der Physik, vol. 38, Issue 11, pp

13 Kef laio 2. Uposthrizìmenoi RujmoÐ sth Diepif neia Met llou DihlektrikoÔ se meg lo eôroc suqnot twn mporeð na perigrafeð me to montèlo pl smatoc (plasma model). Sto montèlo pl smatoc h dom enìc met llou mporeð na prosomoiwjeð wc èna nèfoc hlektronðwn pou kineðtai mèsa se èna perib llon jetik fortismènwn pur nwn. Me b sh to montèlo autì h dihlektrik stajer twn met llwn apoteleð sun rthsh mìno thc suqnìthtac kai exart tai apì paramètrouc pou kajorðzontai apì to eðdoc tou met llou. 'Etsi isqôei, ωp 2 ε(ω) =1 ω 2 + jγω, (2.1) ìpou ω p, h suqnìthta pl smatoc tou met llou (p nw apì thn opoða to mètallo q nei tic hlektromagnhtikèc metallikèc tou idiìthtec) kai γ =1/τ h suqnìthta sugkroôsewn twn hlektronðwn. To parap nw montèlo perigr fei ikanopoihtik th sumperifor twn perissìterwn met llwn èwc kai arket uyhlèc suqnìthtec (p.q gia ta alk lia to montèlo isqôei èwc kai thn upèrujrh perioq ), en antðjeta den perigr fei tìso swst th sumperifor twn eugen n met llwn (qrusìc, rguroc) kont stic optikèc suqnìthtec. Gia to l ìgo autì prot jhkan mia seir llwn montèlwn, praktik epekt sewn tou montèlou Drude, shmantikìtero apì ta opoða apoteleð to montèlo Lorentz Drude. SÔmfwna me autì to montèlo h dihlektrik stajer twn met llwn perigr fetai apì thn akìloujh sqèsh, ìpou kai ε r (ω) =ε (f) (ω)+ε (b) (ω) (2.2) ε (b) r = Ω 2 ε r (f) p =1 ω(ω jγ ) k i=1 (2.3) f i ω 2 p (ω 2 i ω2 )+jωγ i, (2.4) ìpou ω p eðnai h suqnìthta pl smatoc tou met llou, k o arijmìc twn talantwt n me suqnìthta ω i, dônamh f i kai qrìno zw c 1/Γ i, en Ω p = f o ω p h suqnìthta pl smatoc pou sqetðzetai me tic metab seic entìc twn stoib dwn, antðstoiqh dônamh f o kai suqnìthta aporrìfhshc Γ o. Oi par metroi pou emfanðzontai stic parap nw exis seic exart ntai apì to eðdoc tou met llou kai mporoôn na brejoôn stic sqetikèc ergasðec twn Johnson kai Rakic [5, 6]. 2.2 Majhmatik perigraf twn rujm n H exagwg thc analutik c èkfrashc twn rujm n pou uposthrðzontai sth diepif - neia met llou peperasmènhc agwgimìthtac kai dihlektrikoô, prokôptei apì thn epðlush twn exis sewn tou Maxwell gia th gewmetrða pou faðnetai sto sq ma 2.1 Upojètontac armonik metabol twn pediak n megej n sunart sei tou qrìnou ( / t jω) oi exis seic tou Maxwell mporoôn na graftoôn wc ex c: E = jωμ H, H = jωɛe. (2.5a ) (2.5b ) 7

14 Kef laio 2. Uposthrizìmenoi RujmoÐ sth Diepif neia Met llou DihlektrikoÔ διηλεκτρικό z μέταλλο y x (διεύθυνση διάδοσης) Sq ma 2.1: Basik gewmetrða diepif neiac met llou dihlektrikoô: h di dosh gðnetai kat th dieôjunsh x en to epðpedo z =sumpðptei me thn epif neia epaf c twn dôo ulik n. Apì ton sundiasmì twn parap nw dôo exis sewn katal goume sth gnwst exðswsh Helmholtz 2 E + k 2 E =, (2.6) ìpou k = ω/c. Gia thn epðlush tou sugkekrimènou probl matoc èqoume th disdi stath gewmetrða pou faðnetai kai sto sq ma 2.2. H di dosh tou rujmoô gðnetai kat th dieôjunsh x, h dihlektrik stajer lamb nei tic diakritèc timèc ε 1 = ε dihlektrikoô gia z>kai ε 2 (ω) =ε met llou (ω) gia z<kai tèloc lamb noume wc dedomèno ìti den up rqei kamða metabol sthn k jeth dieôjunsh y ( / y ) Me b sh tic parap nw paradoqèc to hlektrikì pedðo tou kômatoc gr fetai, Antikajist ntac thn Ex. (2.7) sthn Ex. (2.6) prokôptei E(x, y, z) =E(z)e jβx (2.7) 2 E(z) z 2 +(k 2 ε β 2 )E(z) = (2.8) z x 1 Διηλεκτρικό 2 Μέταλλο Sq ma 2.2: Disdi stath apeikìnish thc gewmetrðac thc diepif neiac met llou dihlektrikoô 8

15 Kef laio 2. Uposthrizìmenoi RujmoÐ sth Diepif neia Met llou DihlektrikoÔ Epistrèfoume stic Ex. (2.5) tic opoðec analôoume gia k je mða sunist sa. E z y E y z = jωμ H x, (2.9a ) E x z E z x = jωμ H y, (2.9b ) E y x E x y = jωμ H z, (2.9g ) H z y H y z = jωεe x, (2.9d ) H x z H z x = jωεe y, (2.9e ) H y x H x y = jωεe z. (2.9± ) Stic Ex. (2.9) antikajhstoôme ìti / y kai / x = jβ, katal gontac, E y z = jωμ H x E x z + jβe z = jωμ H y jβe y = jωμ H z (2.1a ) (2.1b ) (2.1g ) H y z = jωεe x (2.1d ) H x z + jβh z = jωεe y (2.1e ) jβh y = jωεe z (2.1± ) Apì tic parap nw exis seic mporoôme na ex goume dôo diaforetik sônola lôsewn pou antistoiqoôn stouc TE (E z =) kai TM rujmoôc (H z =), antðstoiqa. TM rujmoð Gia thn perðptwsh twn rujm n ìpou H z =to sônol otwn exis sewn 2.1 gðnetai, E x = j H y ωε z E z = β ωε H y (2.11a ) (2.11b ) kai upojètontac lôsh gia to H y ekjetik apsbenômmenh kaj c apomakrunìmaste apì to epðpedo z =gia tic dôo perioqèc, èqoume tic ekfr seic, 9

16 Kef laio 2. Uposthrizìmenoi RujmoÐ sth Diepif neia Met llou DihlektrikoÔ gia z> H y = A 1 e k 1z e jβx k 1 E x = ja 1 e k1z e jβx ωε ε 1 β E z = A 1 e k1z e jβx ωε ε 1 (2.12a ) (2.12b ) (2.12g ) gia z< H y = A 2 e k 2z e jβx k 2 E x = ja 2 e k2z e jβx ωε ε 2 β E z = A 2 e k2z e jβx ωε ε 2 (2.13a ) (2.13b ) (2.13g ) Shmei noume ìti stic Ex. (2.12) kai (2.13) oi ε 1 kai ε 2 eðnai oi sqetikèc dihlektrikèc stajerèc twn hmiq rwn 1 kai 2 antðstoiqa. Efarmìzontac th sunèqeia twn efaptomenik n sunistws n tou magnhtikoô kai hlektrikoô pedðou sthn epif neia z =dhlad H y1 z= = H y2 z= kai E x1 z= = E x2 z= prokôptei ìti A 1 = A 2 kai epðshc k 2 k 1 = ε 2 ε 1. (2.14) Prèpei na shmei soume gia thn Ex. (2.14) ìti èqoume upojèsei pwc Re{k 1 } > kai Re{k 2 } > prokeimènou h lôsh na èqei fusik shmasða. 'Etsi, sthn parap nw exðswsh parathroôme ìti prokeimènou na ikanopoieðtai h sunj kh sunèqeiac prèpei ta ε 1 kai ε 2 na eðnai eterìshma, apaðthsh pou ikanopoieðtai kaj c sômfwna me to montèlo tou Drude ta mètalla emfanðzoun arnhtikì pragmatikì mèroc thc sqetik c dihlektrik c stajer c touc gia suqnìthtec mikrìterec thc suqnìthtac pl smatoc. Pèra apì ta parap nw, h èkfrash thc H y prèpei na ikanopoieð kai thn exðswsh Helmholtz, ètsi antikajist ntac thn èkfrash thc H y gia k je mða apì tic dôo perioqèc sthn Ex. (2.8) katal goume sto parak tw sônolo exis sewn, k 2 1 = β2 k 2 ε 1 k 2 2 = β2 k 2 ε 2 (2.15a ) (2.15b ) apì tic opoðec se sundiasmì me thn Ex. (2.14) katal goume sth sqèsh diaspor c gia touc rujmoôc SPP pou uposthrðzontai apì th diepif neia met llou dihlektrikoô. β = k ε1 ε 2 ε 1 + ε 2 (2.16) Sthn parap nw sqèsh diaspor c, h opoða isqôei tìso gia pragmatikèc ìso kai migadikèc timèc thc ε 2, h suqnìthta emplèketai tautìqrona sto k kai sto ε 2 (ω) dhmiourg ntac mia mh grammik sqèsh diaspor c gia touc rujmoôc SPP. 1

17 Kef laio 2. Uposthrizìmenoi RujmoÐ sth Diepif neia Met llou DihlektrikoÔ TE rujmoð Sthn perðptwsh twn TE rujm n (E z =)apì to sônolo twn Ex. (2.1), upojètontac kai p li gia thn E y lôsh parìmoia thc perðptwshc twn TM rujm n èqoume tic akìloujec ekfr seic gia tic dôo perioqèc, gia z> E y = A 1 e k 1z e jβx k 1 H x = ja 1 e k1z e jβx ωμ β H z = A 1 e k1z e jβx ωμ (2.17a ) (2.17b ) (2.17g ) gia z< E y = A 2 e k 2z e jβx k 2 H x = ja 2 e k2z e jβx ωμ β H z = A 2 e k2z e jβx ωμ (2.18a ) (2.18b ) (2.18g ) Efarmìzontac th sunèqeia twn efaptomenik n sunistws n tou magnhtikoô kai hlektrikoô pedðou sthn epif neia z =dhlad E y1 z= = E y2 z= kai H x1 z= = H x2 z= prokôptei ìti A 1 = A 2 kai epðshc A 1 (k 1 + k 2 )=. (2.19) 'Omwc prokeimènou h arqik lôsh pou upojèsame gia thn E y na èqei fusikì nìhma prèpei E y ìtan z ± ra Re{k 1 } > kai Re{k 2 } >. Prokeimènou dhlad na isqôoun oi oriakèc sunj kec katal goume sto sumpèrasma ìti A 1 = A 2 =. To parap nw apotèlesma apodeiknôei thn adunamða upost rixhc TE SPP rujm n sth diepif neia met llou dihlektrikoô. 2.3 ProsomoÐwsh EpibebaÐwsh twn SPP rujm n Met thn exagwg thc analutik c èkfrashc twn SPP rujm n pou uposthrðzontai sth gewmetrða pou melet same, mporoôme na sugkrðnoume ta analutik apotelesmatame ta apotelèsmata thc prosomoðwshc me th bo jeia thc mejìdou peperasmènwn stoiqeðwn (Finite Elements Method, FEM). Gia thn epðlush tou sugkekrimènou probl matoc me th mèjodo peperasmènwn stoiqeðwn qrhsimopoi jhke to emporik diajèsimo pakèto logismikoô Comsol 3.4 kai h exagwg twn diagramm twn gia tic analutikèc exis seic ègine me th bo jeia tou Matlab. 11

18 Kef laio 2. Uposthrizìmenoi RujmoÐ sth Diepif neia Met llou DihlektrikoÔ Sq ma 2.3: Q rthc tou pl touc thc sunist sac H y glass. Sth diaqwristik Au silica DiakrÐnetai emfan c h apìsbesh tou pl touc kaj c o rujmìc diadðdetai proc ta dexi kai tautìqrona h meðwsh tou profðltou rujmoô kaj c apomakrunìmaste apì th diaqwristik epif neia. Prin proqwr soume sthn parousðash kai sqoliasmì twn diagr mmatwn ja asqolhjoôme lðgo perissìtero me th morf kai tic idiìthtec twn rujm n SPP. H sqèsh gia thn sunist sa H y eðnai, H y = A 1 e k iz e jβx (2.2) β 2 kε 2 1 gia i =1,z> ìpou k i = kai β = k ε1 ε 2 β2 kε 2 ε 1 +ε 2. 2 gia i =2,z< 'Etsi me dedomèno ìti h sqetik dihlektrik stajer twn met llwn sômfwna me to montèlo tou Drude eðnai migadikìc arijmìc, mporoôme na gr youme th stajer di doshc c β = Re{β} + jim{β}. Sthn parap nw sqèsh to pragmatikì mèroc antistoiqeð sth fasik stajer di doshc en to fantastikì mèroc eis gei tic ap leiec pou emfanðzei to pl toc tou rujmoô kat th di dosh tou. Sto sq ma 2.3 ta apotelèsmata thc prosomoðwshc di taxhc qrusoô (Au) kai silica glass sto m koc kômatoc 1.31 mm. To m koc thc di taxhc eðnai 5mm, to p qoc tou dihlektrikoô 5mm kai to p qoc tou met llou 18nm. O deðkthc di jlashc tou qrusoô se autì to m koc kômatoc isoôtai me n Au =.41 j8.42 kai o deðkthc di jlashc tou dihlektrikoô Sto sq ma 2.3 diakrðnoume thn ekjetik apìsbesh pou emfanðzei to pl toc tou pedðou kaj c o rujmìc diadðdetai proc ta dexi kai tautìqrona th meðwsh tou pl touc kaj c apomakrunìmaste apì thn epif neia epaf c twn dôo ulik n. ParathroÔme dhlad ìti o rujmìc emfanðzei th mègisth sugkèntrwsh tou akrib c sthn diepif neia met llou dihlektrikoô. To teleutaðo autì sumpèrasma faðnetai kalôtera sto sq ma 2.4 ìpou parathroôme thn ekjetik meðwsh pou emfanðzei to profðltou rujmoô kaj c apomakrunìmaste apì thn epif neia epaf c. H sugkèntrwsh tou rujmoô gðnetai kurðwc sto dihlektrikì 12

19 Kef laio 2. Uposthrizìmenoi RujmoÐ sth Diepif neia Met llou DihlektrikoÔ 1 Magnetic Field norm [A/m] Magnetic Field norm [A/m] y -6 1 (m) Sq ma 2.4: ProfÐl thc magnhtik c pediak c èntashc sto x =. π Αναλυτική angle(h ) (rad) y -π Αναλυτική x (διεύθυνση διάδοσης) (αʹ) Πλάτος της H y συνιστώσας x (διεύθυνση διάδοσης) (βʹ) Φάσητης H y συνιστώσας Sq ma 2.5: Apeikìnish tou pl touc kai thc f shc thc H y sunist sac kat m koc thc dieôjunshc di doshc, me kìkkino parousi zontai ta analutik apotelèsmata, en me mplè ta apotelèsmata thc prosomoðwshc me to Comsol. 13

20 Kef laio 2. Uposthrizìmenoi RujmoÐ sth Diepif neia Met llou DihlektrikoÔ Κυκλική συχνότητα ω /ω p air silica ω sp,air ω sp,silica 1 Σταθερά διάδοσης βc/ω p Sq ma 2.6: Di gramma diaspor c SPP rujm n. Me suneqeðc grammèc eðnai to pragmatikì mèroc tou β, me diakekomènec to fantastikì, en oi eujeðec grammèc antistoiqoôn stic eujeðec fwtìc tou k je dihlektrikoô kai perigr - fontai apì th sqèsh k = ω c εd ìpou ε d h sqetik dihlektrik tou dihlektrikoô se k je perðptwsh. kaj c ìpwc parathroôme to profðl fjðnei polô gr gora mèsa sto mètallo. Sth sunèqeia ja sugkrðnoume ta apotelèsmata pou proèkuyan apì thn prosomoðwsh thc di taxhc me th bo jeia tou Comsol me ta apotelèsmata pou ex qjhsan kat thn analutik epðlush tou probl matoc. Sta sq mata 2.5 parathroôme ìti ta apotelèsmata thc prosomoðwshc dðnoun apotelèsmata pou tautðzontai me ta analutik apotelèsmata, gegonìc pou mac epitrèpei na jewr soume ìti h prosomoðwsh twn diat xewn aut n me th mèjodo twn peperasmènwn stoiqeðwn eðnai ikanopoihtik. Shmei noume gia llh mia for thn ekjetik apìsbesh tou pl touc kai th grammik metabol thc f shc pou parousi zei o rujmìc kat m koc thc di doshc. 2.4 Di gramma diaspor c SPP rujm n Met thn analutik exagwg thc morf c twn rujm n pou uposthrðzontai apì thn mon diepif neia met llou dihlektrikoô mporoôme na melet soume kai na sqoli soume to di gramma diaspor c twn rujm n. H melèth ja perioristeð sthn perðptwsh twn TM rujm n, kaj c ìpwc apodeðqjhke kai parap nw, oi TE rujmoð den uposthrðzontai sth gewmetrða aut. H morf thc kampôlhc diaspor c prokôptei apì thn Ex. (2.16) antikajist ntac thn analutik èkfrash thc dihlektrik c stajer c tou met llou sunart sei thc suqnìthtac. 'UpenjumÐzoume kai p li ìti analutikèc ekfr seic thc dihlektrik c stajer c twn shmantikìterwn met llwn sugkentr nontai stic ergasðec twn Rakic kai Johnson [5, 6]. Sto sq ma 2.6 parousðazontai ta diagr mmata diaspor c gia dôo diaforetikèc gewmetrðec met llou dihlektrikoô (mètallo/aèrac, mètallo/sio 2 ). H dihlektrik stajer tou met llou prokôptei apì to montèlo tou Drude jewr ntac mhdenik suqnìthta sugkroôsewn hlektronðwn kai dôo diaforetik dihlektrik, aèra kai silica glass. Lìgw 14

21 Kef laio 2. Uposthrizìmenoi RujmoÐ sth Diepif neia Met llou DihlektrikoÔ thc morf c twn rujm n SPP, suzeugmènoi sth diepif neia met llou dihlektrikoô, h kampôlh diaspor c touc brðsketai sta dexi thc gramm sc fwtìc, perioq sthn opoða ja esti soume to endi fèron mac. ParathroÔme pr ton ìti, gia suqnìthtec megalôterec thc suqnìthtac pl smatoc tou met llou, ìpou to mètallo q nei tic hlektromagnhtikèc idiìthtec tou, o rujmìc mporeð kai aktinoboleðtai, kai deôteron ìti up rqei mia z nh suqnot twn ìpou to b paðrnei mìno fantastikèc timèc apagoreôntac thn di dosh tou rujmoô se ekeðnh thn perioq dhmiourg ntac ousiastik mia apagoreumènh z nh suqnot twn (bandgap). H kampôlh diaspor c twn suzeugmènwn rujm n SPP stic qamhlèc suqnìthtec akoloujeð thn kampôlh tou fwtìc kai epekteðnetai gia poll m kh kômatoc sto dihlektrikì dhmiourg ntac ta kômata pou eðnai gnwst wc Sommerfeld-Zenneck kômata. AntÐjeta, ìso h stajer di doshc megal nei, h suqnìthta teðnei asumptwtik se mða suqnìthta, pou sumbolðzetai wc ω sp. Gia thn perðptwsh met llou sto opoðo upojèsame montèlo Drude qwrðc ap leiec Im[ε 2 (ω)] = prokôptei, ω sp = ω p 1+ε1 (2.21) 'Otan to b teðnei sto peiro h taqôthta om dac tou rujmoô, u g, teðnei sto mhdèn (u g ), dhmiourg ntac ètsi ènan mh diadidìmeno all kai mh aktinoboloômeno rujmì me kajar hlektrostatik sumperifor. H sumperifor aut perigr fetai wc Surface Plasmon. 2.5 Diègersh SPP rujm n sth diepif neia met llou dihlektrikoô Oi rujmoð SPP pou diadðdontai kat m koc thc diepif neiac met llou dihlektrikoô eðnai praktik, hlektromagnhtik kômata suzeugmèna sthn epif neia epaf c twn dôo ulik n. H sugkèntrwsh tou rujmoô epitugq netai, kaj c h stajer di doshc β eðnai megalôterh tou kumatikoô arijmoô k sto dihlektrikì, odhg ntac se ekjetik aposbenômmeno profðl tou rujmoô ìso apomakrunìmaste apì thn epif neia epaf c. 'Opwc analôjhke kai sthn prohgoômenh enìthta, h kampôlh diaspor c twn SPP rujm n brðsketai sta dexi thc gramm c fwtìc tou dihlektrikoô. H morf kai h jèsh thc kampôlhc diaspor c upodeiknôoun ton trìpo me ton opoðo mporeð na gðnei h sôzeuxh fwtìc kai SPP rujmoô. H jèsh thc kampôlhc diaspor c twn SPP rujm n, dexi thc gramm c fwtìc tou dihlektrikoô apokleðei th dunatìthta apeujeðac sôzeuxhc mèsw k poiac prospðtpousac aktðnac fwtìc, kaj c aut se k je perðptwsh ja èprepe na briskìtan sta arister thc gramm c fwtìc tou dihlektrikoô. Sthn enìthta aut ja parousi soume teqnikèc me tic opoðec h parap nw sôzeuxh kajðstatai efikt SÔzeuxh mèsw prðsmatoc Oi rujmoð SPP den mporoôn na diegerjoôn apeujeðac me aktðnec fwtìc kaj c gia th stajer di doshc isqôei β > k ìpou k h stajer di doshc (kumatikìc arijmìc) tou fwtìc sto dihlektrikì. 'Etsi h probol tou kumatikoô dianôsmatoc k sthn orizìntia 15

22 Kef laio 2. Uposthrizìmenoi RujmoÐ sth Diepif neia Met llou DihlektrikoÔ αέρας πρίσμα διεπιφάνεια μετάλλου / αέρα Συχνότητα διεπιφάνεια μετάλλου / πρίσματος Κυματικό διάνυσμα στην οριζόντια διεύθυνση Sq ma 2.7: SÔzeuxh mèsw prðsmatoc kai di gramma diaspor c SPP rujm n. Mìno stajerèc di doshc pou brðskontai an mesa stic dôo grammèc fwtìc mporoôn na epiteuqjoôn. dieôjunsh (dieôjunsh di doshc) k x = k sin θ (ìpou θ h gwnða prìsptwshc tou fwtìc metroômenh wc proc thn katakìrufo) eðnai p nta mikrìterh thc stajer c di doshc β twn SPP rujm n akìma kai sthn oriak perðptwsh thc sqedìn orizìntiac prìsptwshc (grazing incidence). Par' ìla aut, diègersh SPP rujm n mèsw thc prosarmog c f shc (phase matching) mporeð na epiteuqjeð se èna sôsthma tri n epipèdwn pou apoteleðtai apì èna leptì str ma met llou an mesa se dôo dihlektrik diaforetikoô deðkth di jlashc. Gia aplìthta ja jewr soume wc èna apì ta dihlektrik ton aèra(ε 1 =1). Mia aktðna fwtìc pou diadðdetai sto dihlektrikì uyhlìterou deðkth di jlashc (ε 2 ) kai anakl tai sthn epif neia met llou dihlektrikoô ja èqei stajer di doshc sto orizìntio epðpedo k x = k ε2 sin θ (k hstajer di doshc ston aèra, ε 2 hdihlektrik stajer tou uperstr matoc/prðsmatoc, θ h gwnða prìsptwshc wc proc thn katakìrufo thc epif neiac) pou epitrèpei th sôzeuxh me SPP rujmì, all sth diepif neia met llou-aèra. H idèa aut apotup netai kalôtera sto sq ma 2.7. Analutikìtera, ta diadidìmena kômata entìc tou dihlektrikoô megalôterou deðkth di jlashc èqoun stajer di doshc sthn orizìntia dieôjunsh, k x, pou brðsketai arister thc gramm c fwtìc tou dihlektrikoô. 'Osa apì aut ta kômata èqoun orizìntia stajer di doshc an mesa sthn gramm fwtìc tou dihlektrikoô kai tou aèra mporoôn na diegeðroun SPP rujmoôc sth diepif neia met llou aèra. H mèjodoc aut gnwst kai wc exasjenhmènh olik eswterik an klash (attenuated total internal reflection) perilamb nei th dièleush (tunneling) tou pedðou thc diègershc sthn perioq met llou aèra, ìpou gðnetai h sôzeuxh me ton antðstoiqo SPP rujmì. Gia thn ulopoðhsh thc sôzeuxhc fwtìc SPP rujm n èqoun protajeð dôo diaforetikèc gewmetrðec, ìpwc faðnetai kai sto sq ma 2.8. H pio sun jhc mèjodoc eðnai h teqnik Kretschmann Raether 2, sq ma 2.8(a ), sthn opoða èna leptì fôllo met l- 2 Η τεχνική παρουσιάστηκε από τους Kretschmann και Raether το 1968 στην εργασία Radiative 16

23 Kef laio 2. Uposthrizìmenoi RujmoÐ sth Diepif neia Met llou DihlektrikoÔ (αʹ) (βʹ) Sq ma 2.8: Diaforetikèc gewmetrðec sôzeuxhc me th qr sh prðsmatoc. (a') UlopoÐhsh Kretschmann Raether ìpou parathroôme to fainìmeno thc dieôleshc tou pedðou mèsw tou met llou kai th sôzeuxh tou rujmoô sth diepif neia met llou aèra kai (b') ulopoðhsh Otto ìpou diakrðnetai to leptì str ma aèra metaxô prðsmatoc kai met llou. lou enapotðjetai p nw se èna gu llino prðsma. Ta fwtìnia thc optik c aktðnac pou proskroôoun sth diepif neia prðsmatoc met llou me gwnða megalôterh thc krðsimhc èqoun th dunatìthta na dièljoun mèsw tou met llou kai na suzeuqjoôn me SPP rujmoôc sth diepif neia met llou aèra. Mia diaforetik prosèggish apoteleð h mèjodoc Otto 3, sq ma 2.8(b ), sthn opoða to mètallo diaqwrðzetai apì to prðsma me èna leptì str ma aèra SÔzeuxh me th qr sh fr gmatoc par jlashc anaglôfou H diafor twn kumatik n dianusm twn, thc orizìntiac sunist sac k x = k sin θ tou prospðptontoc kômatoc kai thc stajer c di doshc β, mporeð na kalufjeð me thn kat llhlh periodik diamìrfwsh thc epif neiac tou met llou me aul kia opèc, dhlad me èna fr gma par jlashc. Se aut thn perðptwsh h periodik di taxh 'prosfèrei' to kumatikì di nusma sth dieôjunsh di doshc pou apaiteðtai ste to prospðpton na suzeuqjeð me ton SPP rujmì. Prokeimènou na up rxei h prosarmog f shc prèpei na isqôei, β = k sin θ ± nk G (2.22) ìpou k G =2π/a to amoibaðo di nusma thc periodik c di taxhc kai n =(1, 2, 3,...). EÐnai emfanèc apì ta parap nw ìti h periodik di taxh mporeð na qrhsimopoihjeð kai gia thn antðstrofh diadikasða, dhlad gia thn aposôzeuxh tou SPP rujmoô se optikì kuma pou diadðdetai sto dihlektrikì apomakrunìmeno apì th diaqwristik. M lista ì- pwc parathroôme kai apì thn Ex. (2.22) mporoôme na elègxoume th gwnða aposôzeuxhc decay of nonradiative surface plasmons excited by light, Z. Naturforsch. Teil A 23, (1968). 3 Η μέθοδος αυτή παρουσιάστηκε επίσης το 1968 όπως και η μέθοδος Kretschmann Raether, στην εργασία Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection, Z. Phys. 216, (1968). 17

24 Kef laio 2. Uposthrizìmenoi RujmoÐ sth Diepif neia Met llou DihlektrikoÔ k k a Sq ma 2.9: Prosarmog f shc tou fwtìc me SPP rujmì me th qr sh fr gmatoc par jlashc me allag twn gewmetrik n qarakthristik n thc periodik c di taxhc (all zontac praktik thn perðodo thc). Profan c to fr gma par jlashc eðnai h aploôsterh enallaktik, kaj c parousi zei periodikìthta sth mða di stash. 'Enac fwtonikìc krôstalloc prosfèrei an logh sumperifor me perissìterouc bajmoôc eleujerðac. Prosomoi seic pou èqoun gðnei stic diat xeic aposôzeuxhc [7] upodeiknôoun th dunatìthta aposôzeuxhc èwc kai 5% thc eiserqìmenhc isqôoc. Perissìterec plhroforðec gia thn epðdrash thc gewmetrðac thc periodik c dom c/fr gmatoc par jlashc stic i- diìthtec twn diat xewn sôzeuxhc kai aposôzeuxhc mporoôn na brejoôn sth didaktorik diatrib tou Alexey Krasavin [7]. Peiramatikèc ergasðec sqetik me thn sôzeuxh kai aposôzeuxh me th qr sh periodik n dom n mporoôn brejoôn stic ergasðec twn Park kai Devaux [8, 9]. 18

25 Kef laio 3 Metaôlik arnhtikoô deðkth di jlashc 3.1 Genik perð metaôlik n arnhtikoô deðkth di - jlashc Ta metaôlik (metamaterials) eðnai ulik sta opoða oi fusikèc idiìthtec prokôptoun apì th dom kai ìqi apì th fusik touc sôstash. Gia na ta diakrðnoume apì lla sônjeta ulik, qarakthrðzoume telik wc metaôlik aut ta opoða emfanðzoun asun jistec fusikèc idiìthtec. O ìroc eis qjh to 1999 apì ton Rodger M. Walser sto University of Texas at Austin. Sta plaðsia thc hlektromagnhtik c jewrðac, ta metaôlik parousi zoun exairetikì endiafèron kaj c mporoôn na qrhsimopoihjoôn se mða plhj ra efarmog n, ìpwc tèleioi fakoð, diamorfwtèc, mikrokumatikoð zeôktec kai kalômmata gia keraðec (antenna radomes). Ta metaôlik ephre zoun ta hlektromagnhtik kômata, parousi zontac ìmwc metabol twn gewmetrik n qarakthristik n kai thc dom c touc se klðmaka mikrìterh apì to m koc kômatoc tou pedðou me to opoðo allhlepidroôn. 'Etsi mporoôn na jewrhjoôn praktik, wc omoiogen ulik se ìlh touc thn èktash. O deðkthc di jlashc, o opoðoc eðnai to pio jemeli dec mègejoc gia na perigr youme thn allhlepðdrash thc hlektromagnhtik c aktinobolðac me thn Ôlh, eðnai genik migadikìc arijmìc, n = n + jn, ìpou to n jewreðtai jetikìc. Parìlo pou h sunj kh n < de parabi zei kanèna jemeli dh fusikì nìmo, ulik pou parousi zoun arnhtikì pragmatikì mèroc tou deðkth di jlashc emfanðzoun asun jistec idðothtec. Gia par deigma, to fwc pou prospðptei sth diaqwristik epif neia metaxô dôo mèswn me jetikì kai arnhtikì pragmatikì mèroc tou deðkth di jlashc, diajl tai me arnhtik gwnða wc proc thn katakìrufo en tautìqrona kat thn di dosh tou sto mèso me arnhtikì deðkth di jlashc, h fasik taqôthta kai h taqôthta om dac eðnai antipar llhlec, ìpwc kai to kumatikì di nusma me to di nusma Poynting, en ta dianôsmata E, H, k sqhmatðzoun èna aristerìstrofo sôsthma. Ex' aitðac twn parap nw idiot twn, ta ulik aut apokaloôntai aristerìstrofa ulik (left handed materials) metaôlik arnhtikoô deðkth di jlashc (negative index materials). H jemelðwsh thc jewrðac thc arnhtik c fasik c taqôthtac ra kai twn aristerìstrofwn ulik n ègine kurðwc apì ton Veselago [1]. 19

26 Kef laio 3. Metaôlik arnhtikoô deðkth di jlashc α β γ z x y Sq ma 3.1: (a) Magnhtik suntonizìmeno stoiqeðo, (b) Hlektrik suntonizìmeno stoiqeðo, (g) o sundiasmìc touc mporeð na dhmiourg sei ulikì me n < Genik, aristerìstrofa ulik den up rqoun sth fôsh, me k poiec sp niec exairèseic ìpwc to bismoôjio, to opoðo emfanðzei n < gia λ 6mm. Par' ìla aut den eðnai gnwstì k poio fusikì ulikì to opoðo na parousi zei arnhtikì deðkth di jlashc stic optikèc suqnìthtec. 'Etsi eðnai aparaðthto na strafoôme se teqnht ulik, pou eðnai kataskeuasmèna kat tètoio trìpo, ste na parousi zoun sunolik arnhtikì deðkth di jlashc. 'Opwc èqoume anafèrei kai parap nw, mac endiafèroun ulik ta opoða parousi zoun periodikìthta thc dom c touc se klðmaka mikrìterh tou m kouc kômatoc, ste na mporoôme na ta antimetwpðsoume wc isodônama omoiogen ulik me sugkekrimèna qarakthristik se ìlh thn èktash touc. Mia pijan all ìqi kai monadik prosèggish prokeimènou na epitôgqoume arnhtikì deðkth di jlashc, eðnai na sqedi soume èna ulikì sto opoðo h isotropik dihlektrik stajer ε = ε +jε kai h isotropik magnhtik diaperatìthta μ = μ +jμ upakoôoun sthn exðswsh, ε μ + μ ε <. (3.1) Autì odhgeð se arnhtikì pragmatikì mèroc tou deðkth di jlashc, ìpwc apodeðqjhke apì touc Depine, Lakhtakia [11]. H Ex. (3.1) ikanopoieðtai gia ε < kai μ <. Prèpei na shmei soume ìmwc ìti to teleutaðo sumpèrasma den eðnai anagkaða sunj kh, kaj c mporeð na up rqoun energ magnhtik ulik (μ 1), me jetikì pragmatikì mèroc thc magnhtik c diaperatìthtac, gia ta opoða h Ex. (3.1) ikanopoieðtai kai ra isqôei ìti n <. H pr th prosèggish gia to p c ja kataskeuastoôn energ magnhtik ulik, prot jhke apì ton Pendry to 1999 [12]. Duo omìkentroi anoiqtoð daktôlioi, oi opoðoi eðnai prosanatolismènoi se antðjetec dieujônseic kai èqoun mègejoc mikrìtero tou m kouc kômatoc, problèfjhke ìti mporoôn na odhg soun se μ <. O sundiasmìc twn parap nw ulik n me gnwst metaôlik pou emfanðzoun arnhtik dihlektrik stajer stic suqnìthtec melèthc thc prohgoômenhc di taxhc, dhlad sta 1GHz, dhmioôrghse to pr to metaôlikì me tautìqrona arnhtikì pragmatikì mèroc thc dihlektrik c stajer c kai thc magnhtik c diaperatìthtac kai kat sunèpeia arnhtikì pragmatikì mèroc 2

27 Kef laio 3. Metaôlik arnhtikoô deðkth di jlashc Sq ma 3.2: Di taxh par llhlwn metallik n nanor bdwn. Sto sq ma parousi zetai kai h kat llhlh pìlwsh pou prèpei na èqei to fwc prokeimènou h di taxh na emfanðzei n <. tou deðkth di jlashc, ìpwc parousi sthke sthn ergasða twn Smith, Schultz [13]. H di taxh parousi zetai sto sq ma 3.1. Apì tìte h prosp jeia gia th metafor aut c thc sumperifor c se uyhlìterec suqnìthtec basðsthke arqik sth smðkrunsh twn diat xewn me apotèlesma thn emf nish arnhtik c dihlektrik c stajer c mèqri kai to 1THz. H lôsh prokeimènou na epitôgqoume arnhtikì deðkth di jlashc stic optikèc suqnìthtec, eðnai na ekmetalleutoôme to fainìmeno twn plasmonik n rujm n pou parousi - zontai sthn epif neia met llwn, ìpwc o qrusìc kai o rguroc stic suqnìthtec autèc. Wc gnwstìn stic optikèc suqnìthtec ta mètalla aut èqoun arnhtikì pragmatikì mèroc th dihlektrik c stajer c. O sundiasmìc autoô tou fainomènou me k poia kat llhlh gewmetrða pou ja parousi zei energ magnhtik sumperifor kai m lista μ < se k poio eôroc suqnot twn, ìpou ε <, eðnai ikan sunj kh ste na èqoume èna ulikì me arnhtikì pragmatikì mèroc tou deðkth di jlashc stic optikèc suqnìthtec. H parap nw zhtoômenh gewmetrða brèjhke ìti mporeð na eðnai mða periodik di taxh sthn opoða to monadiaðo kelð apoteleðtai apì dôo par llhlec metallikèc nanor bdouc ìpwc faðnetai sto sq ma 3.2. H jewrhtik ergasða twn Podolskiy, Sarychev, Shalev [14], upèdeixe th dunatìthta emf nishc diamagnhtik c apìkrishc aut n twn diat xewn sta 15nm. Autì to apotèlesma, se sundiasmì me touc plasmonikoôc rujmoôc kai thn arnhtik dihlektrik stajer pou parousi zei h di taxh sto dedomèno m koc kômatoc, od ghse sthn prìbleyh ìti ulik pou perièqoun tètoiec par llhlec nanor bdouc mporoôn na emfanðsoun arnhtikì pragmatikì mèroc tou deðkth di jlashc akìma kai stic optikèc suqnìthtec. H peiramatik epibebaðwsh tou arnhtikoô deðkth di jlashc, gia to shmantikì apì thlepikoinwniak c pleur c m koc kômatoc twn 15nm, ègine apì ton Shalaev [15]. H di taxh gia thn opoða o arnhtikìc deðkthc di jlashc epeteôqjh, parousi zetai sto sq ma 3.3. H di taxh apoteleðtai apì metallikèc nanor bdouc p qouc 5nm pou eðnai topojethmènec se gu lino upìstrwma kai diaqwrðzontai metaxô touc apì str ma dihlektrikoô SiO 2 p qouc 6nm. Sthn pragmatikìthta h r bdoc sthn koruf thc di taxhc èqei mikrìterec diast seic apì thn r bdo thc b shc sqhmatðzontac mia puramidik gewmetrða. H dom pou perigr fhke, epanalamb netai periodik stic dôo diast seic prokeimènou na dhmiourghjeð to metaôlikì. 21

28 Kef laio 3. Metaôlik arnhtikoô deðkth di jlashc (α) β δ γ Sq ma 3.3: (a)apì arister proc ta dexi, di taxh metallik n nanor bdwn kai kat llhlh pìlwsh fwtìc gia thn emf nish arnhtikoô deðkth di jlashc, eikìna hlektronikoô mikroskopðou kai diast seic, (b) Periodik di taxh nanor bdwn (g) apeikìnish hlektronikoô mikroskopðou thc periodik c di taxhc (d) diast seic monadiaðou kelioô periodik c di taxhc. 22

29 Kef laio 3. Metaôlik arnhtikoô deðkth di jlashc 3.2 Mèjodoc prosdiorismoô energoô deðkth di - jlashc n eff Oi klðmakec diast sewn periodikìthtac kai dom c twn metaôlik n, tèjhke wc proôpìjesh na eðnai mikrìterec apì to m koc kômatoc tou hlektromagnhtikoô pedðou me to opoðo ja allhlepidr soun. H idiìthta aut mac epitrèpei na antimetwpðzoume th di taxh wc mða isodônamh omoiogen di taxh me stajer fusik qarakthristik se ìlh thn èktash thc. H prosèggish aut, ìpou th jèsh tou anomoiogenoôc ulikoô paðrnei èna isodônamo omoiogenèc ulikì me Ðdia makroskopik sumperifor, apokaleðtai prosèggish isodônamou mèsou (effective medium approach) kai eðnai idiaðtera qr simh gia th melèth kai ton qarakthrismì thc sumperifor c twn metaôlik n. Sta plaðsia aut c thc prosèggishc, h dunatìthta na upologðsoume ton isodônamo deðkth diajlashc pou parousi zoun autèc oi periodikèc diat xeic, prosfèrei mia pl rh eikìna gia ton trìpo pou allhlepidroôn oi diat xeic me thn hlektromagnhtik aktinobolða. 'Enac trìpoc gia na prosdiorðsoume ton energì deðkth di jlashc thc di taxhc, prokôptei me th bo jeia tou nìmou tou Snell, metr ntac th gwnða di jlashc tou prospðptontoc kômatoc. An parathr soume ìmwc th di taxh tou Shalaev, aut èqei sunolikì p qoc mìlic 16nm, kajist ntac th mèjodo aut praktik adônath. EÐnai aparaðthth loipìn, mia diaforetik mèjodoc gia ton prosdiorismì tou energoô deðkth di jlashc me th qr sh megej n, sthn tim twn opoðwn èqoume mesh prìsbash eðte peiramatik eðte mèsw prosomoðwshc. Mia tètoia mèjodoc prot jhke apì touc Soukoulis, Markos, Smith [4], sthn opoða o upologismìc tou energoô deðkth di jlashc thc di taxhc gðnetai mèsw twn suntelest n an klashc kai met doshc. Sto sq ma 3.4 parousi zetai h di taxh sthn opoða epðpedo kôma prospðptei sth diepif neia metaxô aèra kai tou ulikoô, tou opoðou ton energì deðkth di jlashc jèloume na prosdiorðsoume. 'Ena mèroc tou kômatoc anakl tai en èna llo diadðdetai. H di taxh mporeð na jewrhjeð me b sh th jewrða mikrokum twn, wc èna dðjuro sto opoðo o suntelest c an klashc, r, antistoiqeð ston ìro S 11 en o suntelest c met doshc, t, ston ìro S 21 tou pðnakac skèdashc [S]. OpÐnakac metafor c thc sugkekrimènhc di taxhc eðnai, [ ] [ ] A B cos (neff kd) jz sin (n eff kd) = C D j 1 z sin (n (3.2) effkd) cos(n eff kd) Qrhsimopoi ntac tic gnwstèc apì th jewrða mikrokum twn, sqèseic metasqhmatismoô metaxô twn pin kwn metafor kai skèdashc èqoume r S 11 = t S 21 = j (z z 1 )sin(n eff kd) cos (n eff kd)+ j 2 (z + z 1 )sin(n eff kd) 1 cos (n eff kd)+ j 2 (z + z 1 )sin(n eff kd) (3.3a ) (3.3b ) EpilÔontac tic Ex. (3.3a ), (3.3b ) wcproc ta n eff,z èqoume, cos (n eff kd) = 1 2t [ 1 ( r 2 t 2)] (3.4) 23

30 Kef laio 3. Metaôlik arnhtikoô deðkth di jlashc d n eff, z n=1 n =1 Προσπίπτον r t air metamaterial air Sq ma 3.4: Di taxh an klashc met doshc gia ton upologismì tou energoô deðkth di jlashc me th qr sh sqèsewn metasqhmatismoô tou pðnaka metafor c se pðnaka skèdashc. kai z = ± (1 + r) 2 t 2 (1 r) 2 t 2 (3.5) ParathroÔme, ìti en genik oi ekfr seic gia ta n eff,z eðnai aplèc, eðnai migadikèc sunart seic me pollaploôc kl douc, gegonìc pou mporeð na dhmiourg sei as feia wc proc thn epilog tou swstoô kl dou thc lôshc. Prokeimènou na xeperastoôn autèc oi as feiec, epikaloômaste mia seir fusik n periorism n. 'Etsi sthn perðptwsh mh energ n ulik n prèpei Re{z} >, ra sthn Ex. (3.5) epilègetai ekeðnh h rðza pou dðnei jetikì pragmatikì mèroc. Me thn Ðdia logik, gia mh energ ulik, prèpei to fantastikì mèroc tou energoô deðkth di jlashc na eðnai mikrìtero tou mhdenìc (upenjumðzoume ìti autì isqôei gia thn sômbash twn mhqanik n / t jω). 'Etsi gia ton energì deðkth di jlashc èqoume, ] [ 1 cos 1 2t (1 r2 + t 2 ) Re{n eff } = ±Re kd [ ] 1 cos 1 2t (1 r2 + t 2 ) Im{n eff } = ±Im kd + 2πm kd (3.6a ) (3.6b ) kai sthn Ex. (3.6b ) epilegoume th rðza pou dðnei arnhtikì fantastikì mèroc gia to n eff. 'Eqontac xekajarðsei to jèma tou pros mou, h epilog gia to pragmatikì mèroc prafan c upakoôei sthn epilog pou ègine gia to fantastikì, mènei na deukrinðsoume poion apì touc kl douc thc Ex. (3.6a ) ja epilèxoume. Gia meg lec timèc tou p qouc tou ulikoô, d, parathroôme ìti oi kl doi thc Ex. (3.6a ) keðtontai polô kont metaxô touc kajist ntac dôskolh thn epilog tou swstoô kl dou. EÐnai profanèc ìti ta kalôtera apotelèsmata epitugq nontai gia polô mikrèc timèc tou p qouc thc di taxhc. Me skopì na xeperasteð aut h duskolða ston prosdiorismì tou swstoô kl dou, proteðnetai na 24

31 Kef laio 3. Metaôlik arnhtikoô deðkth di jlashc Χρυσός SiO 2 (αʹ) (βʹ) Sq ma 3.5: (a) Trisdi stath eikìna di taxhc metaôlikoô pou parousi zei arnhtikì pragmatikì mèroc tou deðkth di jlashc kai (b) monadiaðo kelð thc di taxhc. gðnetai mètrhsh gia dôo perissìtera p qh thc di taxhc prokeimènou na epilegoôn oi swstoð kl doi pou dðnoun k je for to Ðdio apotèlesma. 3.3 ProsomoÐwsh di taxhc arnhtikoô deðkth di - jlashc Met ton kajorismì thc mejìdou pou akoloujeðtai gia ton prosdiorismì tou energoô deðkth di jlashc, ja proqwr soume se aut n thn enìthta sthn prosomoðwsh miac periodik c di taxhc, h opoða ìpwc ja apodeiqjeð emfanðzei arnhtikì pragmatikì mèroc tou energoô deðkth di jlashc se sugkekrimèna m kh kômatoc. Mia apeikìnish thc di - taxhc pou ja prosomoiwjeð parousi zetai sto sq ma3.5(a ). H di taxh apoteleðtai apì zeôgh metallik n fôllwn qrusoô pou ekteðnontai kat thn dieôjunsh y, diaqwrðzontai metaxô touc me dihlektrikì kai epanalamb nontai periodik kat th dieôjunsh x. H tom thc di taxhc sto epðpedo xz parousi zetai sto sq ma 3.5(b ). Ta dôo metallik fôlla èqoun p qoc d, to sunolikì p qoc thc di taxhc eðnai D 1, to pl toc twn metallik n fôllwn eðnai w kai h perðodoc thc di taxhc eðnai h. Tèloc na shmei soume ìti h pìlwsh tou prospðptontoc fwtìc eðnai TM (sunist sa magnhtikoô pedðou kat m koc tou xona y) kaj c se aut thn pìlwsh parathr jhke h emf nish arnhtikoô deðkth di jlashc. H prosomoðwsh thc di taxhc ja gðnei me th mèjodo peperasmènwn stoiqeðwn, me th bo jeia tou pakètou logismikoô Comsol v.3.4. UpenjumÐzoume ìti gia na prosdiorðsoume ton energì deðkth di jlashc me th mèjodo twn Soukoulis, Markos, Smith qreiazìmaste mìno touc suntelestèc an klashc kai met doshc thc di taxhc. Sta plað- sia thc prosomoðwshc ja jewr soume mða perioq aèra prin to metaôlikì, mða perioq aèra met kai mia perioq PML sto tèloc thc gewmetrðac prokeimènou na apofôgoume tic anakl seic. To str ma tèleiac prosarmog c (Perfectly Matched Layer, PML) eðnai èna teqnhtì str ma apwlei n, pou suqn qrhsimopoieðtai kat thn epðlush twn kumatik n exis sewn me th mèjodo peperasmènwn diafor n sto pedðo tou qrìnou (Finite Differences Time Domain, FDTD) methmèjodo peperasmènwn stoiqeðwn (Finite E- 25

32 Kef laio 3. Metaôlik arnhtikoô deðkth di jlashc lement Method, FEM). H basik idiìthta tou PML, pou to xeqwrðzei apì ta upìloipa ulik me ap leiec eðnai ìti, èqei sqediasteð kat tètoio trìpo, ste èna epðpedo kôma pou prospðtei sthn epif neia tou PML apì k poio llo fusikì mèso den anakl tai, all diadðdetai sto PML ìpou aposbènetai polô gr gora. Aut h idiìthta epitrèpei sto PML na aporrof sei to exerqìmeno, apì thn upologistik perioq, hlektromagnhtikì kôma, kai na mhn epitrèyei thn an klash tou proc to eswterikì. Stic eikìnec pou ja parousiastoôn apì thn prosomoðwsh, h perioq tou PML de ja emfanðzetai kaj c den antistoiqeð se fusikì q ro. Me th qr sh tou Comsol, mporoôme na èqoume mesh prìsbash sthn tim thc sunist sac H y se k je shmeðo thc di taxhc. 'Etsi ja proqwr soume ston upologismì twn suntelest n an klashc kai met doshc arijmhtik afoô isqôei, ( ) A H A y Hy inc,a dx r = (3.7) H inc,a y dx kai t = B HB y dx A Hinc,A y dx, (3.8) ìpou ta A kai B anafèrontai stic perioqèc stic opoðec orðzoume ton suntelest an - klashc kai met doshc. Prìkeitai gia th diepif neia epaf c thc perioq c aèra me to metaôlikì apì thn pleur thc prìsptwshc kai th diepif neia epaf c tou aèra me to metaôlikì sthn perioq thc met doshc antðstoiqa. Stic parap nw sqèseic emfanðzetai mìno h H y sunist sa kaj c eðnai kai h mình mh mhdenik sunist sa tou magnhtikoô pedðou. H prosomoðwsh ja gðnei gia dôo diaforetikèc diat xeic qrusoô kai SiO 2 stic opoðec ja diathrhjoôn stajer ìla ta megèjh ektìc apì to p qoc twn metallik n fôllwn. Gia thn pr th prosomoðwsh èqoume w = 45nm, h = 48nm, D 1 =16nm kai d=19nm. H prosomoðwsh gðnetai jewr ntac gia th dihlektrik stajer tou qrusoô to montèlo Drude kai to montèlo Lorentz Drude, sômfwna me ìsa èqoun perigrafeð sto kef laio 1. Oi par metroi twn sqèsewn pou perigr foun th dihlektrik stajer tou qrusoô sunart sei tou m kouc kômatoc el fjhsan apì thn ergasða tou Rakic [6]. H prosomoðwsh ja gðnei gia ta m kh kômatoc apì 7 èwc 13nm, me metablhtì b ma, prokeimènou na èqoume epark an lush sta endi mesa m kh kômatoc. Sto sq ma 3.6(a ) parousi zetai to pl toc thc H y sunist sac gia to m kockômatoc twn 9nm, en sta sq mata 3.6(b ) kai 3.6(g ) to pragmatikì mèroc tou deðkth di jlashc ìpwc prokôptei me th mèjodo twn Soukoulis, Markos, Smith pou perigr - fhke prohgoumènwc, kai lamb nontac ta dôo diaforetik montèla gia thn dihlektrik stajer tou qrusoô. ParathroÔme ìti se auto to p qoc twn metallik n fôllwn den emfanðzetai poujen arnhtikì pragmatikì mèroc tou deðkth di jlashc. Sth deôterh prosomoðwsh, mei noume to p qoc twn metallik n fôllwn se d=13nm en diathroôme ìlec tic upìloipec diast seic stajerèc. 'Etsi gia th deôterh prosomoðwsh èqoume sto sq ma 3.7(a ) thn apeikìnish tou pl touc thc H y sunist sac ìpwc prokôptei apì to Comsol gia to m koc kômatoc twn 85nm kai sta sq mata 3.7(b ), 3.7(g ) to pragmatikì mèroc tou deðkth di jlashc ìpwc prokôptei apì th mèjodo twn Soukoulis, Markos, Smith gia ta dôo diaforetik montèla pou perigr foun th dihlektrik stajer tou qrusoô. 26

33 Kef laio 3. Metaôlik arnhtikoô deðkth di jlashc Re{n } eff (αʹ) Πλάτος H y συνιστώσας σε όλη την έκταση της διάταξης για λ=9nm λ (m) x 1-6 (βʹ) Πραγματικό μέρος του n eff με τηχρήση μοντέλου Drude Re{n } eff λ (m) x1-6 (γʹ) Πραγματικό μέρος του n eff με τηχρήση μοντέλου Lorentz Drude Sq ma 3.6: Par metroi prosomoðwshc, w = 45nm, h = 48nm, D 1 =16nm, d=19nm. To (a) eðnai gia to m koc kômatoc twn 9nm, en sta (b) kai (g) apeikonðzetai to pragmatikì mèroc tou deðkth di jlashc me th qr sh diaforetik n montèlwn gia th dihlektrik diaspor tou qrusoô. Gia tic parap nw paramètrouc den parathreðtai arnhtikì n se kanèna m koc kômatoc. 27

34 Kef laio 3. Metaôlik arnhtikoô deðkth di jlashc Re{n } eff (αʹ) Πλάτος H y συνιστώσας σε όλη την έκταση της διάταξης για λ=85nm λ (m) x1-6 (βʹ) Πραγματικό μέρος του n eff με τηχρήση μοντέλου Drude Re{n } eff λ (m) x1-6 (γʹ) Πραγματικό μέρος του n eff με τηχρήσημοντέλου Lorentz Drude Sq ma 3.7: Par metroi prosomoðwshc, w = 45nm, h = 48nm, D 1 =16nm, d=13nm.to (a) eðnai gia to m koc kômatoc twn 85nm, en ta (b) kai (g) epeikonðzoun to pragmatikì mèroc tou deðkth di jlashc me th qr sh diaforetik n montèlwn gia th dihlektrik diaspor tou qrusoô. ParathroÔme ìti kai me ta dôo diajèsima montèla problèpetai arnhtikì pragmatikì mèroc tou deðkth di jlashc gia m kh kômatoc gôrw apì ta 85nm. 28

35 Kef laio 3. Metaôlik arnhtikoô deðkth di jlashc ParathroÔme ìti se aut thn perðptwsh, kai gia ta dôo montèla diaspor c thc dihlektrik c stajer c tou qrusoô, up rqei èna eôroc mhk n kômatoc sto opoðo to pragmatikì mèroc tou deðkth di jlashc lamb nei arnhtikèc timèc. To el qisto emfanðzetai kai tic dôo forèc konta sta 85nm. Genik, prèpei na shmei soume ìti h qr sh twn dôo diaforetik n montèlwn, Drude kai Lorentz Drude, diaforopoieð shmantik ta apotelèsmata, upodeiknôontac ìti k je mða apì autèc tic prosomoi seic prèpei na epibebaiwjeð kai peiramatik ste plèon me bebaiìthta na mporeð na eipwjeð ìti k - poia apì tic diat xeic autèc parousi zei arnhtikì deðkth di jlashc stic anaferìmenec suqnìthtec. H parap nw di taxh èqei prosomoiwjeð sta plaðsia rjrou tou A.V Kildishev [16] kai ta apotelèsmata pou paratðjontai se aut thn enìthta parousi zoun polô kal sumfwnða me ta antðstoiqa dhmosieumèna. 29

36 Kef laio 4 Mèjodoc Prosarmog c Rujm n AntikeÐmeno tou kefalaðou autoô, apoteleð h analutik diaqeðrish thc allhlepðdrashc periodik diamorfwmènwn gewmetri n met llwn kai hlektromagnhtik n kum twn, me th qr sh thc mejìdou prosarmog c rujm n (Mode Matching Method). H mèjodoc aut basðzetai sta jewr mata twn Floquet kai Bloch, sômfwna me ta opoða k je hlektromagnhtikì kôma pou diadðdetai se èna periodikì mèso, mporeð na grafteð wc èna peiro jroisma diakrit n epðpedwn kum twn, to kaj' èna apì ta opoða diadðdetai se mia austhr orismènh dieôjunsh. Sundi zei de ta parap nw, me thn efarmog twn oriak n sunjhk n tou Maxwell stic diaqwristikèc epif neiec kai th st jmish twn exis sewn pou prokôptoun me k poio kat llhlo eswterikì ginìmeno, ìpwc ja faneð analutikìtera parak tw. 4.1 Genikèc idiìthtec epðpedwn kum twn An lush epðpedou kômatoc se s kai p pìlwsh Prin xekin soume thn an lush thc mejìdou prosarmog c rujm n, ja exet soume k poiec basikèc idiìthtec twn epðpedwn kum twn. Wc gnwstìn k je epðpedo omoiìmorfo kôma pou diadðdetai entìc kajorismènou trisorjog niou sust matoc suntetagmènwn, mporeð na analujeð sto dianusmatikì jroisma thc s kai p pìlwshc. H s pìlwsh jewreðtai aut, sthn opoða to di nusma tou hlektrikoô pedðou eðnai k jeto sto epðpedo pou sqhmatðzoun h katakìrufh dieôjunsh sthn epif neia prìsptwshc kai h dieôjunsh tou kumatikoô dianôsmatoc tou epðpedou kômatoc, en p, h pìlwsh sthn opoða to hlektrikì pedðo eðnai k jeto sth dieôjunsh di doshc kai entìc tou epipèdou pou orðzei to kumatikì di nusma kai h k jetoc sthn epif neia prìsptwshc. Mia apeikìnish thc parap nw perigraf c emfanðzetai sto sq ma 4.1. 'Etsi sthn perðptwsh thc prìsptwshc me azimoujiak gwnða φ inc kai gwnða zenðj θ inc to kumatikì di nusma mporeð na grafteð wc, k inc = k xˆx + k y ŷ + k z ẑ (4.1) 3

37 Kef laio 4. Mèjodoc Prosarmog c Rujm n p inc inc k inc s x y z Sq ma 4.1: Trisorjog nio sôsthma suntetagmènwn kai epðpedo kôma pou prospðptei sto epðpedo xy upì tuqaða dieôjunsh. EmfanÐzontai ta dianôsmata twn s kai p pol sewn, kaj c kai to kumatikì di nusma thc di doshc. ìpou, k x = k sin θ inc cos φ inc, k y = k sin θ inc sin φ inc, k z = k cos θ inc. (4.2a ) (4.2b ) (4.2g ) EÐnai shmantikì na ekfr soume k je mða perðptwsh pìlwshc se kartesianèc sunist sec. 'Etsi gia k je èna apì ta dianôsmata k inc, s kai p èqoume, k inc = k x k y k z k y s = k x p = k x k y k2 x + k2 y k z (4.3a ) (4.3b ) (4.3g ) 'Opwc faðnetai kai apì tic Ex.(4.3) ta dianôsmata s kai p ikanopoioôn tic sunj kec pou tèjhkan prohgoumènwc, dhlad sqhmatðzoun me to k inc èna trisorjog nio sôsthma suntetagmènwn, to s eðnai k jeto sto epðpedo pou orðzoun ta k inc kai ẑ kai to p brðsketai entìc tou epipèdou autoô. 31

38 Kef laio 4. Mèjodoc Prosarmog c Rujm n 'Etsi èqoume gia k je mða pìlwsh to antðstoiqo hlektrikì pedðo. [ ] ky E s = A s e jkxx jkyy jkzz en (4.4a ) k x [ ] kx E p = A p e jkxx jkyy jkzz (4.4b ) k y ìpou parap nw emfanðzontai mìno oi x kai y sunist sec, kaj c eðnai autèc pou parousi zoun endiafèron kai ja mac apasqol soun sthn efarmog twn oriak n sunjhk n, ìpwc ja faneð kai parak tw. Sthn perðptwsh thc katakìrufhc di doshc, dhlad gia k inc = k z ẑ h epilog an mesa sth s kai p pìlwsh eðnai aujaðreth, opìte sta pl aðsia aut c thc ergasðac epilègoume s = [ ] 1 p = [ ] 1 kai to hlektrikì pedðo antðstoiqa eðnai, [ ] [ ] 1 E s = A s e jkzz E 1 p = A p e jkzz An lush rujm n Floquet Bloch SÔmfwna me to je rhma Floquet Bloch, k je kôma pou prokôptei wc an klash enìc epðpedou kômatoc me k inc = k xˆx + k yŷ + k zẑ apì mða periodik di taxh, mporeð na anaptuqjeð se èna peiro jroisma epðpedwn kum twn kaj' èna apì ta opoða diadðdetai se mia orismènh dieôjunsh me ènan dikì tou suntelest an klashc. H dieôjunsh di doshc kaj' enìc apì aut ta epðpeda kômata prokôptei apì to je rhma Floquet. EpÐshc prèpei na prosjèsoume ìti, se k jemia apì autèc tic diakritèc dieujônseic di doshc enup rqoun tautìqrona kai h s kai h p pìlwsh. UpenjumÐzoume ìti h an lush thc s kai p pìlwshc se kartesianèc sunist sec gðnetai monos manta an dieôjunsh di doshc ìpwc exhg jhke sthn prohgoômenh par grafo. 'Ara mporoôme na gr youme ìti E ref = (A nm s ŝ nm + A nm p ˆp nm )e jkn x x jkm y y+jknm z z (4.5) nm ìpou kx n = k x + 2πn L x = k y + 2πm k m y k nm z = L y k 2 sup (kn x )2 ( k m y (4.6a ) (4.6b ) ) 2 (4.6g ) 32

39 Kef laio 4. Mèjodoc Prosarmog c Rujm n p L x Ly k nm inc inc kinc s x y Sq ma 4.2: An logo tou sq matoc 4.1, ed parousi zetai epðshc kai h periodikìthta kat x kai y kaj c kai èna sônolo parajl menwn rujm n Floquet. z gia n, m =,..., kai ŝ nm = ˆp nm = [ ] 1 k m y [ L x L y (kx n)2 + ( ) ] ky m 2 kx n (4.7a ) [ ] 1 k n x [ L x L y (kx n)2 + ( ) ] ky m 2 ky m. (4.7b ) Prèpei epðshc na shmei soume ed ìti, sthn perðptwsh pou to parajl meno kôma diadðdetai katakìrufa dhlad kx n = km y =tìte, [ ] 1 ŝ nm = Lx L y 1 ˆp nm = [ ] 1 1. Lx L y 'Etsi mporoôme na xanagr youme ta parap nw, eis gontac tautìqrona ton sumbolismì pou ja qrhsimopoihjeð sth sunèqeia kat thn epðlush twn problhm twn. Genik to hlektrikì pedðo ja sqetðzetai me 3 diaforetikèc paramètrouc, to n, to m kai to σ, apì ta opoða, to pr to kajorðzei ton ìro Floquet kat thn x dieôjunsh, to deôtero ton ìro Floquet kat thn y dieôjunsh kai tèloc to σ kajorðzei to eðdoc thc pìlwshc s p. 'Etsi, [ ] E E nm nm σ = x,σ Ey,σ nm exp( jkx n x jkm y y ± jknm z z) (4.8) ìpou gia n = m =kai epilègontac to ston ìro tou jkz nm z anaferìmaste sto 33

40 Kef laio 4. Mèjodoc Prosarmog c Rujm n prospðpton kôma, en k je lloc ìroc me + eðnai profan c ìroc tou anakl menou kai [ ] 1 k n x [ [ ] L E nm x L y (kx n)2 + ( ) ] ky m 2 k m gia σ = p y x,σ Ey,σ nm = [ ] (4.9) 1 k m y [ L x L y (kx n)2 + ( ) ] ky m 2 kx n gia σ = s en gia thn periptwsh pou kx n = ky m di doshc, [ ] E nm x,σ Ey,σ nm = =dhlad gia thn perðptwsh thc katakìrufhc [ ] 1 1 Lx L y [ ] 1 Lx L y 1 gia σ = p gia σ = s. (4.1) EÐnai shmantikì na shmei soume se autì to shmeðo k poiec idiìthtec pou fèroun oi rujmoð Floquet, oi opoðec ja qrhsimopoihjoôn parak tw. 1. Oi rujmoð Floquet eðnai orjog nioi metaxô touc, dhlad Ly 2 Ly 2 Lx 2 Lx 2 E nm σ (E n m σ ) dxdy = gia n n,m m kai σ σ (4.11) 2. To eswterikì ginìmeno k je rujmoô me ton eautì tou dðnei mon da. Ly 2 Ly 2 Lx 2 Lx 2 E nm σ 2 dxdy =1 (4.12) Exagwg sunistws n magnhtikoô pedðou apì tic ekfr seic tou hlektrikoô pedðou 'Opwc ja deðxoume se aut thn par grafo, oi ekfr seic twn sunistws n kat x kai y tou magnhtikoô pedðou mporoôn na prokôyoun monos manta gia k je perðptwsh pìlwshc, apì tic antðstoiqec ekfr seic tou hlektrikoô pedðou, sundèontac tautìqrona to pedðo me tic ekfr seic pou prokôptoun gia touc rujmoôc Floquet. Sthn perðptwsh tìso thc s, ìso kai thc p pìlwshc to hlektrikì pedðo E gr fetai, E = [ Ex E y ] = [ ] E nm x,σ Ey,σ nm exp( jkx n x jkm y y jknm z z). 34

41 Kef laio 4. Mèjodoc Prosarmog c Rujm n Gia thn perðptwsh thc s pìlwshc to hlektrikì pedðo èqei sunist sec mìno kat x kai y ra apì thn exðswsh strof c tou hlektrikoô pedðou èqoume, E = jωμh, H x = j E y ωμ z = z knm ωμ E y H y = j E x ωμ z = knm z ωμ E x (4.13a ) (4.13b ) Gia thn perðptwsh thc p pìlwshc to magnhtikì pedðo prèpei p li na eðnai k jeto sto hlektrikì all kai sto kumatikì di nusma, ra se aut n thn perðptwsh to H èqei sunist sec mìno kat x kai y. 'Etsi apì thn exðswsh strof c tou magnhtikoô pedðou, ra H = jωεe H x z = jωεe y (4.14a ) H y z = jωεe x (4.14b ) (4.14g ) apì ìpou prokôptei, H x = ωε E kz nm y H y = ωε E x. kz nm (4.15a ) (4.15b ) Sunolik katal xame ìti, oi sunist sec kata x kai y tou magnhtikoô pedðou sundèontai mìno me tic antðstoiqec sunist sec tou hlektrikoô pedðou kai genik mporoôme na gr youme ìti, ìpou Y nm σ = H = Y nm σ kz nm ωμ ωε kz nm [ ] E nm y,σ Ex,σ nm gia σ = p gia σ = s Efarmog thc mejìdou prosarmog c rujm n (4.16) (4.17) Gia na efarmìsoume th mèjodo prosarmog c rujm n, prèpei na qwristeð h di taxh se diakritèc perioqèc gia k je mða apì tic opoðec ja ekfr soume to sunolikì hlektrikì 35

42 Kef laio 4. Mèjodoc Prosarmog c Rujm n kai magnhtikì pedðo wc sun rthsh gnwst n paramètrwn kai gnwstwn oi opoðec ja anazhthjoôn. Efarmìzontac tic oriakèc sunj kec stic di forec diaqwristikèc epif - neiec kai stajmðzontac tic exis seic pou prokôptoun me kat llhla megèjh k je for, katal goume se èna sônol o exis sewn Ðso me to arijmì twn agn stwn tou sust matoc. H parap nw genik kai Ðswc ìqi tìso saf c perigraf ja apotupwjeð kalôtera stic epìmenec enìthtec, ìpou ja efarmosteð h mèjodoc, tìso se aplèc eidikèc gewmetrðec, ìso kai sto genikìtero prìblhma metallikoô fôllou me orjogwnikèc opèc. 4.2 Efarmog thc mejìdou sthn perðptwsh mon c periodikìthtac Prin proqwr soume sthn efarmog thc mejìdou prosarmog c rujm n stic gewmetrðec me disdi stath periodikìthta, ja epiqeir soume na elègxoume thn akrðbeia thc mejìdou me mia seir aploôsterwn gewmetri n monodi stathc periodikìthtac. H sôgkrish twn apotelesm twn sthn perðptwsh twn gewmetri n me monodi stath periodikìthta mporeð na gðnei me ta apotelèsmata pou prokôptoun me th qr sh thc mejìdou peperasmènwn stoiqeðwn gia tic Ðdiec gewmetrðec. H epðlush miac gewmetrðac dôo diast sewn, dhlad me periodikìthta mìno proc th mða dieôjunsh, me th mèjodo twn peperasmènwn stoiqeðwn eðnai saf c ligìtero apaithtik, apì thn poyh thc apaitoômenhc upologistik c isqôoc, se sqèsh me thn epðlush miac trisdi stathc gewmetrðac. 'Etsi ja perioristoôme ston èlegqo thc mejìdou se autèc tic peript seic. H orjìthta thc mejìdou sthn perðptwsh twn 3 diast sewn tekmaðretai apì thn genikìthta thc. 'Etsi parak tw, ja parousiastoôn oi lôseic gia thn perðptwsh aulaki n (grooves) se metallik epif neia kai h perðptwsh orjogwnik n egkop n se metallik epif neia (slits). Profan c h diafor an mesa stic dôo diat xeic eðnai ìti h pr th eðnai di taxh kajar an klashc, en h deôterh eðnai di taxhc met doshc, dhlad sundèetai kai me to fainìmeno thc exairetik c optik c met doshc. Ta apotelèsmata pou ja sugkrijoôn, ja eðnai diagr mmata sunistws n tou hlektrikoô magnhtikoô pedðou kaj c aut ta megèjh eðnai mesa diajèsima apì th lôsh me th qr sh thc mejìdou peperasmènwn stoiqeðwn. Na shmei soume ìti gia thn epðlush twn problhm twn me th mèjodo twn peperasmenwn stoiqeðwn qrhsimopoi jhke to emporik diajèsimo pakèto logismikoô Comsol v Metallik epif neia me orjogwnik aul kia TM rujmoð Ja xekin soume me thn perðptwsh monodi stathc periodikìthtac dhlad orjogwnik aul kia sthn epif neia met llou. To epðpedo optikì kôma prospðptei sthn epif neia upo gwnða zenðj θ inc kai apoteleðtai mìno apì p pìlwsh, ìpwc faðnetai kai sto sq ma 4.3(a ). To magnhtikì pedðo sthn perioq I, anex rthta apì thn gwnða prìsptwshc, èqei sunist sa mìno kat th dieôjunsh y kai isqôei ìti, H I y = e jk xx jk zz + n ρ n e jkn x x+jk n z z (4.18) 36

43 Kef laio 4. Mèjodoc Prosarmog c Rujm n E E k inc H inc - L/2 -w/2 w/2 L/2 x w d Metal z=d L (αʹ) z (βʹ) Sq ma 4.3: (a) Trisdi stath apeikìnish thc diepif neiac allhlepðdrashc met llou me mon periodikìthta kai epðpedou hlektromagnhtikì kômatoc, (b) tom tou monadiaðou kelioô thc periodik c di taxhc sto epðpedo xz. ìpou, k n x = k x + 2πn L (4.19a ) k n z = k 2 (kn x )2 (4.19b ) gia n =,...,+ kai me b sh thn exðswsh strof c tou magnhtikoô pedðou, h x sunist sa tou hlektrikoô pedðou prokôptei, E I x = k z ωε e jk x x jk z z n k n z ωε ρ ne jkn x x+jkn z z. (4.2) Oi perioqèc entìc twn aulaki n, mporoôn na prosomoiwjoôn wc metallikoð kumatodhgoð peirwn par llhlwn plak n. Kaj c to pedðo sto exwterikì twn aulaki n eðnai polwmèno mìno kat p, sto eswterikì mporoôn emfanistoôn mìno oi TM rujmoð. 'Etsi to pedðo sto eswterikì gr fetai, ìpou kai H II y E II x = M = ( G1,m e jβmz + G 2,m e jβmz) φ m (x) (4.21) m= M m= β m ( G1,m e jβmz G 2,m e jβmz) φ m (x) (4.22) ωε [ πm φ m (x) =cos w β m = k 2 ( x + w )] 2. (4.23) ( mπ ) 2 (4.24) w AkoloÔjwc ja efarmìsoume tic oriakèc sunj kec stic epif neiec z = kai z = d. 37

44 Kef laio 4. Mèjodoc Prosarmog c Rujm n Arqik, apì to mhdenismì thc efaptomenik c sunist sac tou hlektrikoô pedðou sth b sh tou aulakioô (z = d, w 2 x w 2 )isqôei, kai ra oi Ex. (4.21), (4.22) gðnontai, E x z=d =, opìte G 2,m = G 1,m e 2jβmd (4.25) H II y = E II x = M m= M 2G 1,m e jβmd cos [β m (z d)] φ m (x) (4.26) m= 2j β m ωε G 1,me jβmd sin [β m (z d)] φ m (x) (4.27) Efarmìzontac tic oriakèc sunj kec sthn epif neia z =èqoume, H I y z= = H II y z= x < w 2 E I x z= = E II x z= x < w 2 E I x z= = x > w 2 (4.28a ) (4.28b ) (4.28g ) èqoume, e jk x x + n ρ n e jkn x x = m= 2G 1,m e jβmd cos ( β m d) φ m (x) x < w 2 (4.29a ) k z ωε e jk x x n k z ωε e jk x x n kz n ωε ρ ne jkn x x = 2j β m ωε G 1,me jβmd sin ( β m d) φ m (x) m= x < w 2 k n z ωε ρ ne jkn x x = x > w 2 (4.29b ) (4.29g ) Pollaplasi zoume thn Ex. (4.29a ) eswterik me φ m (x)1 noigma tou aulakioô ( x < w 2 ) prokôptei, kai oloklhr nontac sto - S m + n ρ n S nm =2G 1,m e jβmd cos (β m d) R m (4.3) 1 Σημειώνεται ότι ο πολλαπλασιασμός με το συζυγές του φ m (x) γίνεται για να έχουμε το μέτρο του ρυθμού, στησυγκεκριμένηπερίπτωσηόμως που το προφίλ του ρυθμού είναι πραγματικό σε κάθε περίπτωση, ο πολλαπλασιασμός με το συζυγές μέγεθος γίνεται για λόγους συνέπειας ως προς τη λογική της μεθόδου προσαρμογής ρυθμών. 38

45 Kef laio 4. Mèjodoc Prosarmog c Rujm n ìpou, S nm = w 2 e jkn x φm (x)dx (4.31) kai R m = w 2 w 2 w 2 w gia m = φ m (x) 2 dx = w gia m 2 (4.32) AntÐstoiqa pollaplasi zoume thn Ex. (4.29b ) eswterik me e jkn x x (to suzugèc enìc tuqaðou rujmoô Floquet), oloklhr noume kai p li sto noigma tou aulakioô ( x <w/2) kai qrhsimopoi ntac thn Ex. (4.29g ) mporoôme na epekteðnoume ta ìria olokl rwshc tou eswterikoô ginomènou metaxô twn rujm n Floquet se ìlo to monadiaðo kelð ( x <L/2), opìte eðnai dunatìn plèon na qrhsimopoihjeð h idiìthta orjogonikìthtac twn rujm n Floquet kai prokôptei, k z Lδ n k n z Lρ n = M j2β m G 1,m e jβmd sin (β m d) Snm (4.33) m= pou isqôei gia k je n =,...,+. Shmei noume gia tic parap nw exis seic, ìti k name qr sh twn idiot twn pou perigr yame sthn enìthta 4.1.2, dhlad ìti to eswterikì ginìmeno enìc rujmoô me k je llo plhn tou Ðdiou dðnei mhdèn kai ìmoia, oti to eswterikì ginìmeno dôo diaforetik n rujm n entìc tou kumatodhgoô dðnei apotèlesma mhden. Me autìn ton trìpo kat ton pollaplasiasmì thc Ex. (4.29b ) me k je ènan apì touc rujmoôc Floquet paðrnoume mia seir exis sewn pou emplèkoun mìno ton suntelest an klashc tou rujmoô me ton opoðo pollaplasi same. ApaloÐfontac thn metablht G 1,m metaxô twn Ex. (4.3), (4.33) sqhmatðzetai èna sôsthma 2N+1 exis sewn wc proc ρ n ìpou N o arijmìc twn ìrwn Floquet pou lamb - noume up' ìyin kat thn epðlush tou sust matoc. To sôsthma pou dhmiourgeðtai, eðnai èna sôsthma thc morf c A [ρ] =B, A N, N A N, N+1 A N,N ρ N B N A N+1, N A N+1, N+1 A N+1,N ρ N = B N+1 (4.34). A N, N A N, N+1 A N,N ρ N B N ìpou A i,k = j M β m i k kz il tan (β m d) Si,m R S k,m, m= m M A i,k =1+ j i=k kzl i m= β m R m tan (β m d) S i,m 2, B i = δ i 2j M β m kz il tan (β m d) S,m Si,m R. m m= (4.35a ) (4.35b ) (4.35g ) 39

46 Kef laio 4. Mèjodoc Prosarmog c Rujm n gia i, k = N,...,N. AkoloÔjwc, ja efarmosteð h mèjodoc prosarmog c rujm n, ìpwc perigr fhke parap nw, kai ja sugkrijoôn ta exagìmena diagr mmata metaantðstoiqa pou prokôptoun apì thn epðlush tou probl matoc me th qr sh thc mejìdou peperasmènwn stoiqeðwn. Parousi zontai, me th seir pou anafèretai, h metabol tou pl touc thc H y sunist sac se ìlh thn èktash thc gewmetrðac kai sth sunèqeia to Ðdio mègejoc, kat m koc touc xona z sto epðpedo x =. Oi par metroi prosomoðwshc thc pr thc gewmetrðac eðnai suqnìthta, f = 2GHz kai diast seic L =2cm, w =.7cm, d =1cm. Parathr ntac ta diagr mmata twn sqhm twn 4.4 kai 4.5 prokôptei ìti, h sôgklish metaxô twn dôo entel c diaforetik n mejìdwn epðlushc tou probl matoc faðnetai ikanopoihtik. Sto sq ma 4.4 pou apeikonðzetai to pl toc thc y sunist sac tou magnhtikoô pedðou, pou eðnai kai h monadik mh mhdenik, sto epðpedo xz, parathreðtai ìti, h morf tou pedðou sto monadiaðo kelðem- fanðzei meg lh sumfwnða metaxô twn dôo mejìdwn epðlushc, gegonìc pou epibebai nei thn orjìthta thc mejìdou pou akolouj same. Sto sq ma 4.5 prousi zetai mða tom tou pedðou sto epðpedo x =. To di gramma autì ègine me skopì na sugkrðnoume me megalôterh akrðbeia th sôgklish twn dôo mejìdwn. Apì th sôgkrish aut n twn diagramm twn parathreðtai mia mikr apìklish twn tim n, genik ìmwc to mègisto topikì sf lma den xepern to 1-15%, apìklish pou ja jewrhjeð wc apodekt sta plaðsia thc diaforetik c prosèggishc twn dôo mejìdwn. AkoloujoÔn ta diagr mmata pou èginan gia diaforetik di taxh, me perðodo L = 5μm, noigma aulakioô w = 1μm, b joc d = 5μm kai f =1.2THz. Arister se k je sq ma parousi zetai to apotèlesma me th mèjodo peperasmènwn stoiqeðwn kai dexi me th mèjodo prosarmog c rujm n ìpwc kai prohgoumènwc. Se aut thn perðptwsh h sôgklish metaxô twn mejìdwn eðnai saf c kalôterh apì prohgoumènwc, ìpwc faðnetai kai apì ta dôo eðdh diagramm twn. EpÐshc sto sq ma4.6 parathroôme mia sugkèntrwsh tou pedðou gôrw apì to noigma tou aulakioô, stoiqeðo pou parapèmpei amudr se SPP rujmì k ti pou ja exet soume akribèstera sth sunèqeia. 4

47 Kef laio 4. Mèjodoc Prosarmog c Rujm n.1.5 H y z (m) (α ) x (m) (β ).5 Sq ma 4.4: Pl toc thc H y sunist sac: (a') di gramma me th qr sh peperasmènwn stoiqeðwn (b') di gramma me th mèjodo prosarmog c rujm n. Par - metroi gewmetrðac: L =2cm, w =1cm, d =1cmkai f = 2Ghz, h perioq thc prìsptwshc antistoiqeð sta arnhtik z kai stic dôo peript seic. H sôgklish twn mejìdwn eðnai polô ikanopoihtik ìpwc prokôptei kai apì thn sôgkrish twn sqhm twn. To mètro tou dianôsmatoc tou magnhtikoô pedðou thc prìsptwshc eðnai Ðso me th mon da. 3.5 H x= y z (m) Sq ma 4.5: Pl toc thc H y z (m).1 (α ) (β ) sunist sac kat to epðpedo x=, (a') me th mèjodo peperasmènwn stoiqeðwn kai (b') me th mèjodo prosarmog c rujm n. ParathroÔme ìti h apìklish sta apotelèsmata twn dôo diaforetik n mejìdwn de xepern to 1 15%. 41

48 Kef laio 4. Mèjodoc Prosarmog c Rujm n x1-5 6 H [A/m] y z (m) (α ) x (m) x1-5 (α ) Sq ma 4.6: Mètro thc H y sunist sac sth gewmetrða: (a') apotèlesma me th mèjodo peperasmènwn stoiqeðwn (b') apotèlesma me th mèjodo prosarmog c rujm n. Par metroi gewmetrðac L = 5mm, w = 1mm, d = 5mm kai f =1.2THz. ParathroÔme thn polô kal sôgklish pou parousi zoun oi dôo mèjodoi epðlushc. To pl toc thc sunist sac tou magnhtikoô pedðou tou prospðptontoc eðnai Ðso me th mon da. (α ) Η [A/m] x= y z (m) x1-5 (β ) Sq ma 4.7: Pl toc thc H y sunist sac sto epðpedo x=, (a') me th mèjodo peperasmènwn stoiqeðwn kai (b') me th mèjodo prosarmog c rujm n. ParathroÔme ìti h apìklish sta apotelèsmata twn dôo diaforetik n mejìdwn mei jhke se sqèsh me thn perðptwsh twn 2GHz. 42

49 Kef laio 4. Mèjodoc Prosarmog c Rujm n k inc k inc E I II III y w z xx x d inc - L/2 - w/2 w/2 - L/2 x Metal d L (αʹ) z (βʹ) Sq ma 4.8: (a) Trisdi stath apeikìnish thc gewmetrðac metallikoô fôllou me orjogwnik egkop kai (b) monadiaðo kelð thc gewmetrðac Metallik epif neia me orjogwnikèc egkopèc TM rujmoð H perðptwsh di taxhc me orjogwnikèc egkopèc, diafèrei apì thn prohgoômenh gewmetrða, kaj c plèon ta orjogwnik aul kia diapernoôn ìlo to p qoc tou met llou, epitrèpontac th sôzeuxh tou kômatoc sth perioq k tw apì to mètallo. 'Opwc faðnetai kai sto sq ma 4.8(a ) h perioq I eðnai h perioq prìsptwshc tou fwtìc, h perioq II apoteleðtai apì to mètallo kai ta orjogwnik anoðgmata (egkopèc), en h perioq III eðnai h perioq met doshc. GewmetrÐec aut c thc morf c sundèontai me to fainìmeno thc exairetik c optik c met doshc. H kat strwsh kai h epðlush twn exis sewn thc perðptwshc aut c, prokôptei eôkola apì thn prohgoômenh an lush, kaj c h Ðdia diadikasða pou akolouj jhke gia thn epif neia z =prèpei na akoloujhjeð kai gia thn epif neia z = d. 'Etsi pèra apì tic Ex. (4.18), (4.2), (4.21), (4.22) dhlad, H I y = e jk x x jk z z + n ρ n e jkn x x+jkn z z (4.36) E I x = k z ωε e jk x x jk z z n k n z ωε ρ ne jkn x x+jkn z z (4.37) ìpou, H II y M = ( G1,m e jβmz + G 2,m e jβmz) φ m (x) (4.38) m= E II x = M m= β m ( G1,m e jβmz G 2,m e jβmz) φ m (x) (4.39) ωε k n x = k x + 2πn L (4.4a ) k n z = k 2 (kn x )2 (4.4b ) [ πm φ m (x) =cos w ( x + w )] 2 (4.41) 43

50 Kef laio 4. Mèjodoc Prosarmog c Rujm n kai èqoume gia thn perioq III, β m = H y = n k 2 ( mπ ) 2 (4.42) w t n e jkn x jk n z (4.43) E x = n k n z ωε t ne jkn x jkn z. (4.44) Efarmìzontac th sunèqeia twn efaptomenik n sunistws n tou hlektrikoô kai magnhtikoô pedðou stic epif neiec z =kai z = d èqoume, e jk xx + ρ n e jkn xx = (G 1,m + G 2,m ) φ m (x) (4.45a ) n m kz e jk x x kz n ρ ne jkn x = β m (G 1,m G 2,m ) φ m (x) (4.45b ) n m ( G1,m e jβmd + G 2,m e jβmd) φ m (x) = t n e jkn z d e jkn x x (4.45g ) m n ( β m G1,m e jβmd G 2,m e jβmd) φ m (x) = kz n e jkn z d e jkn x x (4.45d ) m n UpenjumÐzetai ìti, sthn perðptwsh thc an lushc twn metallik n epifanei n me orjogwnik aul kia, h exðswsh sunèqeiac tou magnhtikoô pedðou pollaplasi sthke eswterik me to suzugèc k je rujmoô kai to apotèlesma oloklhr jhke sto noigma tou aulakioô ( x <w/2), en h exðswsh sunèqeiac tou hlektrikoô pedðou pollaplasi - sthke me to suzugèc k je rujmoô Floquet kai to apotèlesma oloklhr jhke epðshc sto noigma tou aulakioô. Sthn perðptwsh thc deôterhc exðswshc, qrhsimopoi jhke h exðswsh mhdenismoô tou efaptomenikoô hlektrikoô pedðou sth perioq tou met llou ( x >w/2), me apotèlesma ta ìria olokl rwshc twn rujm n Floquet na epektajoôn se ìlo to monadiaðo kelð, kai ètsi eðnai dunatìn na qrhsimopoihjoôn oi idiìthtec orjogwnikìthtac twn rujm n. H Ðdia akrib c diadikasða akoloujeðtai kai se aut thn perðptwsh, me th diafor ìti Ôparqoun dôo epiplèon exis seic sunèqeiac. ProkÔptei, S m + n ρ n S nm =(G 1,m + G 2,m ) R m (4.46a ) Lk z δ n Lk n z ρ n = m β m (G 1,m G 2,m ) S nm (4.46b ) ( G1,m e jβmd + G 2,m e jβmd) R m = t n e jkn z d S nm n ( β m G1,m e jβmd G 2,m e jβmd) Snm = Lkz n t n e jkn z d m (4.46g ) (4.46d ) 44

51 Kef laio 4. Mèjodoc Prosarmog c Rujm n ìpou, δ i,eðnai to dèlta tou Kronecker kai dðnei mon da mìno gia i =. 2 S nm = w 2 e jkn x x φ m (x)dx (4.47) Jètontac R m = w 2 w 2 w 2 w gia m = φ m (x) 2 dx = w gia m 2 (4.48) A 1,m = β m (G 1,m G 2,m ) ( A 2,m = β m G1,m e jβmd G 2,m e jβmd) (4.49a ) (4.49b ) kai apaloðfontac ta ρ n kai t n katal goume se èna sôsthma wc proc ta A 1,m kai A 2,m : (Σ m G mm ) A 1,m m m A 1,m G mm Σ m A 2,m = I m (4.5a ) Σ m A 1,m +(Σ G mm ) A 2,m m m A 2,m G mm = (4.5b ) Oi posìthtec pou emfanðzontai stic Ex. (4.5) dðnontai apì tic sqèseic Σ m = Σ m = G mm = n G mm = n jr m β m tan (β m d), jr m β m sin (β m d), 1 Lk n z 1 Lk n z (4.51a ) (4.51b ) S nm S nm, (4.51g ) S nm 2, (4.51d ) I m = 2S m (4.51e ) kai k je mða apì tic exis seic isqôei gia k je m =,...,M, dhlad gia k je ènan apì touc rujmoôc entìc tou kumatodhgoô par llhlwn plak n. Oi parap nw exis seic, an loga me ton arijmì twn ìrwn entìc twn aulaki n pou lamb noume upfloyin, sqhmatðzoun èna sôsthma exis sewn pou mporeð na epilujeð me th bo jeia tou upologist. Gia autìn ton lìgo, gr fthke sto Matlab èna prìgramma pou lônei to parap nw sôsthma exis sewn kai upologðzei touc suntelestèc an klashc 2 Προφανώς προκύπτει από τον πολλαπλασιασμό της εξίσωσης συνέχειας του ηλεκτρικού πεδίου στην επιφάνεια z =με τον όρο μηδενικής τάξης, καθώς είναι και ο μόνος όρος που εμφανίζεται και στον όρο ανάκλασης αλλά και στον όρο πρόσπτωσης. 45

52 Kef laio 4. Mèjodoc Prosarmog c Rujm n kai met doshc mèsw twn sqèsewn pou prokôptoun apì tic Ex. (4.46b ), (4.46d ), ìpwc parak tw: ρ n = δ n 1 A Lkz n 1,m Snm, m t n = ejkn z d A Lkz n 2,m Snm. m (4.52a ) (4.52b ) H sôgkrish thc orjìthtac thc mejìdou epðlushc pou parousi sthke, ja gðnei me ta apotelèsmata pou prokôptoun me th mèjodo peperasmènwn stoiqeðwn opwc kai sthn perðptwsh thc metallik c epif neiac me orjogwnik aul kia. Ja parousi soume èna di gramma tou pl touc thc H y sunist sac se ìlh thn èktash tou monadiaðou kelioô, kaj c kai mða tom sto epðpedo x =. Oi diast seic thc di taxhc pou prosomoi netai eðnai, L =2cm, w =.8cm, d =1cmkai h suqnìthta thc prosomoðwshc eðnai f = 2GHz. Sqetik me thn prosomoðwsh, prèpei na upenjumðsoume ìti, jewr same pwc to prospðpton kôma èqei p pìlwsh ra to hlektrikì pedðo èqei sunist sec kat x kai z en to magnhtikì kat y. Oi mìnoi rujmoð pou mporoôn na diegerjoôn ètsi sto eswterikì twn orjogwnik n anoigm twn eðnai oi TM rujmoð. Gia thn epðlush tou sust matoc me th mèjodo thc prosarmog c rujm n l bame up' ìyin, 11 rujmoôc Floquet kai 5 rujmoôc entìc twn orjogwnik n anoigm twn. Sth perðptwsh thc prosmoðwshc thc di taxhc me th mèjodo peperasmènwn stoiqeðwn, pèra apì thn di taxh prostèjhke kai ènac q roc PML met thn perioq thc di doshc prokeimènou na elaqistopoihjoôn oi anakl seic tou kômatoc sth diaqwristik epif neia. O q roc autìc den emfanðzetai sta diagr mmata kaj c den antistoiqeð se fusikì q ro. Parathr ntac to sq ma 4.9 prokôptei ìti, h diadikasða epðlushc sugklðnei, kaj c hmorf tou pedðou se ìlh thn èktash tou monadiaðou kelioô eðnai sômfwnh metaxô twn dôo mejìdwn epðlushc. Sta diagr mmata tou sq matoc 4.1 diapist netai ìmwc mia apìklish stic timèc tou pedðou, parìmoia me th apìklish pou èqei epishmanjeð kai sth melèth twn orjogwnik n aulaki n. Sto sq ma 4.9 h prìsptwsh tou fwtìc eðnai k jeth kai gðnetai apì ta arnhtik z, h perioq pou emfanðzetai qwrðc qr ma (ken ) sto di gramma me th qr sh twn peperasmènwn stoiqeðwn eðnai h perioq tou met llou en antðjeta sto di gramma me th qr sh thc mejìdou prosarmog c rujm n h perioq aut anaparðstai me mplè qr ma. Gia z>1cm briskìmaste sthn perioq thc met doshc. 'Hdh ìmwc prèpei na shmei soume ìti èqoume k poia èndeixh thc Ôparxhc tou fainomènou thc exairetik c optk c met doshc, kaj c sthn perioq thc met doshc h tim tou pedðou lamb nei k poia shmantik tim. 46

53 Kef laio 4. Mèjodoc Prosarmog c Rujm n (αʹ) (βʹ) Sq ma 4.9: Pl toc thc H y sunist sac se ìlh thn èktash tou monadiaðoiu kelioô: (a') di gramma me th qr sh peperasmènwn stoiqeðwn, (b') di gramma me th mèjodo prosarmog c rujm n. Gia th gewmetrða thc prosomoðwshc èqoume, L =2cm, w =.8cm, d =1cmkai f = 2GHz. ParathroÔme thn polô kal sumfwnða twn apotelesm twn pou prokôptoun apì tic dôo diaforetikèc mejìdouc epðlushc. To pl toc tou magnhtikoô pedðou thc prìsptwshc eðnai Ðso me th mon da abs(h ) [A/m] z y (m) (αʹ) H [A/m] x= y z (m) (βʹ) Sq ma 4.1: Pl toc thc H y sunist sac kat m koc tou xona z sto epðpedo x =. (a') Di gramma me th qr sh peperasmènwn stoiqeðwn (b') di gramma me th mèjodo prosarmog c rujm n. ParathroÔme ìti h mègisth topik apìklish metaxô twn dôo mejìdwn den xepern to 1 15% kai ra h sumfwnða twn apotelesm twn krðnetai ikanopoihtik. 47

54 Kef laio 4. Mèjodoc Prosarmog c Rujm n k inc E H d L y I II III w y w x L x Sq ma 4.11: Trisdi stath apeikìnish thc gewmetrða allhlepðdrashc di trhtou met llou kai epðpedou kômatoc. Sto mètallo èqoun anoiqjeð orjogwnikèc opèc periodik, en to f c prospðptei apì thn perioq I. 4.3 Efarmog thc mejìdou sthn perðptwsh dipl c periodikìthtac Sthn enìthta aut, ja parousiasteð h analutik epðlush tou probl matoc thc allhlepðdrashc epðpedou kômatoc, kajorizìmenhc analogðac pol sewn, pou prospðptei upì tuqaða dieôjunsh, se metallikì fôllo sto opoðo èqoun anoiqjeð orjogwnikèc opèc se periodik di taxh. H epðlush ja basisteð sth mèjodo prosarmog c rujm n. H genik gewmetrða thc di taxhc eðnai aut pou faðnetai sto sq ma 4.11, en sto sq ma 4.12 parousi zetai to monadiaðo kelð, sto opoðo ja efarmosteð h an lush. Prokeimènou na epilujeð to prìblhma, ja jewr soume ìti h di taxh qwrðzetai se treðc diakritèc perioqèc I II kai III, ìpwc faðnetai kai sta sq mata 4.11, 4.12, gia k je mða apì tic opoðec, mporoôme na ekfr soume to hlektrikì kai magnhtikì pedðo sunart sei gnwst n paramètrwn kai agn stwn, h tim twn opoðwn ja anazhthjeð. H Ðdia h mejodoc, ìpwc èqoume sqoli sei, sthrðzetai sthn analutik èkfrash twn pedðwn se kaje perioq, sthn efarmog twn oriak n sunjhk n tou hlektrikoô kai magnhtikoô pedðou stic diaqwristikèc epif neiec metaxô twn perioq n kai tèloc sthn epðlush tou grammikoô sust matoc exis sewn pou prokôptei Analutikèc ekfr seic pedðwn Perioq I To pedðo sthn perioq I apoteleð to dianusmatikì jroisma tou prospðptontoc kai tou anakl menou pedðou, opìte gr fetai, E I = E inc + E ref. (4.53) To prospðpton eðnai èna epðpedo kôma pou apoteleðtai apì tuqaða analogða s kai p pol sewn kai diadðdetai se dedomènh dieôjunsh k inc = kxˆx+k yŷ+k z ẑ. EpikaloÔmenoi ton sumbolismì pou jemeli jhke sthn enìthta 4.1.1, to prospðpton kôma sto xy 48

55 Kef laio 4. Mèjodoc Prosarmog c Rujm n epðpedo, mporeð na grafteð wc, ( [ ] E x,s E inc = A s Ey,s [ ]) E x,p + A p Ey,p exp( jkxx jkyy jkz z) (4.54) en to anakl meno, me b sh to je rhma Floquet, prokôptei wc èna peiro jroisma epðpedwn kum twn to kajèna apì ta opoða apoteleðtai me th seir tou apì s all kai apì p pìlwsh. Kat sunèpeia, E ref = n m σ ρ nm σ [ ] E nm x,σ Ey,σ nm exp( jkxx n jky m y + jkz nm z), (4.55) ìpou kx n = k x + 2πn, L x (4.56a ) ky m = k y + 2πm, L y (4.56b ) kz nm = ksup 2 (kn x )2 (ky m)2. (4.56g ) Sunolik gia to pedðo sthn perioq I sundi zontac tic Ex.(4.53),(4.54),(4.55) gr foume, ( [ ] E E I x,s = A s Ey,s + n,m,σ ρ nm σ + A p [ E x,p [ E nm x,σ E nm y,σ Ey,p ] ]) exp( jk x x jk y y jk z z) exp( jk n x x jkm y y + jknm z z), (4.57) ìpou n, m =,..., kai σ = s, p. EpÐshc, me b sh thn an lush pou ègine sthn enìthta 4.1.3, h exagwg twn ekfr sewn tou magnhtikoô pedðou gðnetai apeujeðac apì tic antðstoiqec ekfr seic tou hlektrikoô. Opìte gia to magnhtikì pedðo èqoume, H I = ( A s Y s nmσ [ ] E nm y,s Ex,s nm Yσ nm ρ nm σ + A p Y p [ E nm y,σ E nm x,σ [ ]) E nm y,p Ex,p nm exp( jkxx jkyy jkz z) ] exp( jk n x x jkm y y + jknm z z) (4.58) kai jumðzetai ìti, Y nm σ = ωε k nm z k nm z ωμ gia σ = p gia σ = s (4.59) 49

56 Kef laio 4. Mèjodoc Prosarmog c Rujm n L x Ly wy wx x k inc d I y II z III Sq ma 4.12: MonadiaÐo kelð thc meletoômenhc gewmetrðac. Oi opèc eðnai orjogwnikèc me diast seic w x kai w y, h perðodoc eðnai L x kai L y kat tic x kai y dieujônseic antðstoiqa. To fwc prospðptei upì dieôjunsh k inc. To sôsthma suntetagmènwn eðnai topojethmèno sto kèntro summetrðac twn op n kai sthn epif neia epaf c tou dihlektrikoô tou uperstr matoc me th metallik epif neia. Perioq II H perioq II katalamb netai apì to metallikì str ma sto opoðo èqoun anoiqjeð orjogwnikèc opèc. Se aut thn perioq isqôei ìti, to hlektrikì pedðo eðnai mhdèn sthn perioq tou met llou, en sthn perioq twn op n (oi opoðec mporoôn na prosomoiwjoôn wc orjogwnikoð kumatodhgoð) to pedðo gr fetai wc èna jroisma ìlwn twn dunat n rujm n pou uposthrðzontai. 'Etsi èqoume ìti, E II = gia x > w x 2 kai y > w y 2 (4.6) en gia thn perioq twn op n ( x <w x /2, y <w y /2), [ Ex,α ] (Cα e jβz,αz + D α e jβz,αz ) (4.61) E II = α E y,α ìpou o ìroc α diatrèqei ìlouc touc pijanoôc rujmoôc entìc thc op c, E x,α,e y,α oi sunist sec kat x kai y tou antðstoiqou rujmoô gia k je α, C α,d α oistajerèc pou sqetðzontai me k je ènan rujmì kai th di dos tou kat ta jetik kai arnhtik z antðstoiqa, β z,α hstajer di doshc k je rujmoô. Sthn perðptwsh twn orjogwnik n op n, gia tic sunist sec tou hletrikoô pedðou kai an loga me to eðdoc tou rujmoô, èqoume ta ex c: 5

57 Kef laio 4. Mèjodoc Prosarmog c Rujm n TE mn rujmoð p = mπ w x q = nπ w y (4.62a ) (4.62b ) [ Ex,α E y,α ] = [ ] 2 q cos px sin qy (p2 + q 2 )w x w y p sin px cos qy [ ] 2 q cos px sin qy (p2 + q 2 )w x w y p sin px cos qy an p = q = an p kai q. (4.63) TM mn rujmoð [ Ex,α E y,α ] = p = mπ w x q = nπ w y [ ] 2 p cos px sin qy (p2 + q 2 )w x w y q sin px cos qy (4.64a ) (4.64b ) gia k je p, q (4.65) en, gia k je α isqôei, β z,α = k 2 h p2 q 2 (4.66) ìpou k h = k n h kai n h o deðkthc di jlashc tou ulikoô pl rwshc twn op n. Gia tic sunist sec tou magnhtikoô pedðou, epikaloômenoi kai p li basik stoiqeða thc jewrðac kumatodhg n èqoume, H II = α [ ] Ey,α (Cα Y α e jβz,αz D E α e ) jβz,αz (4.67) x,α ìpou, Y α = β z,α ωμ ωε β z,α TE rujmoð TM rujmoð (4.68) Prèpei kai p li na shmei soume k poiec shmantikèc idiìthtec pou isqôoun gia touc rujmoôc entìc twn op n, an logec me tic idiìthtec pou isqôoun gia touc rujmoôc Floquet. 51

58 Kef laio 4. Mèjodoc Prosarmog c Rujm n To eswterikì ginìmeno k je rujmoô me opoiod pote llo rujmì, eðte TE eðte TM eðnai mhden, wy 2 wy 2 wx 2 wx 2 E α E α dxdy = gia k je α α, (4.69) en to eswterikì ginìmenok jerujmoômeton eautì tou dðnei apotèlesma mon da. wy 2 wy 2 wx 2 wx 2 E α 2 dxdy =1. (4.7) Perioq III Sthn perioq III, emfanðzontai kômata mìno thc morf c rujm n Floquet se dieujônseic Ðdiec me autèc pou eðqame orðsei kai sthn perioq I. Oi suntelestèc k je epðpedou kômatoc eðnai oi gnwstoi pou anazhtoôme kai ja sumbolðzontai parak tw wct nm σ an - loga me to rujmì Floquet kai thn perðptwsh pìlwshc. 'Etsi gr foume, E III = nmσ t nm σ kai antðstoiqa gia to hlektrikì pedðo, H III = nmσ Yσ nm t nm σ [ ] E nm x,σ Ey,σ nm exp( jkx n x jkm y y jknm z z) (4.71) [ ] E nm y,σ Ex,σ nm exp( jkx n x jkm y y jknm z z) (4.72) gia n, m =,..., kai σ = s, p. 3 Efarmog Oriak n Sunjhk n AfoÔ plèon èqoume thn analutik èkfrash tou hlektromagnhtikoô pedðou se k je perioq tou q rou, proqwroôme sthn efarmog twn oriak n sunjhk n tou hlektrikoô kai magnhtikoô pedðou stic epif neiec z =kai z = d. H sunèqeia tou hlektrikoô pedðou epib lletai se ìlh thn èktash tou monadiaðou kelioô,en h sunèqeia tou magnhtikoô 3 Πρέπει να σημειωθεί ότι το kz nm στην περιοχή ΙΙΙ είναι γενικά διαφορετικό του αντίστοιχου της περιοχής Ι καθώς εξαρτάται από τη σταθερά διάδοσης στο διηλεκτρικό. Με δεδομένο ότι σε όλες τις προσομοιώσεις θεωρούμε συμμετρικές διατάξεις με το ίδιο διηλεκτρικό πάνω και κάτω από το διάτρητο μέταλλο, επιλέχθηκε η χρήση του ίδιου συμβόλου για λόγους απλότητας. Σε μια ακόμα πιο γενική προσέγγισηπρέπει να διαχωρίσουμε σε kz,1 nm και knm z,3. 52

59 Kef laio 4. Mèjodoc Prosarmog c Rujm n pedðou mìno sto noigma twn op n. 'Etsi èqoume, E I z= = E II z= H I z= = H II z= E II z=d = E III z=d H II z=d = H III z=d Apì thn Ex. (4.73a ) prokôptei, ( [ ] [ ]) E x,s E A s Ey,s x,p + A p Ey,p e jk x x jk y y + nmσ x < L x 2, y < L y 2 x < w x 2, y < w y 2 x < L x 2, y < L y 2 x < w x 2, y < w y 2 ρ nm σ (4.73a ) (4.73b ) (4.73g ) (4.73d ) [ ] E nm x,σ Ey,σ nm e jkn x x jkm y y = (4.74) kai ( [ ] E x,s A s Ey,s = α + A p [ E x,p [ Ex,α E y,α Ey,p x > w x 2, y > w y 2 ]) e jk x x jk y y + ρ nm σ nmσ ] (C α + D α ) x < w x 2, y < w y 2 [ ] E nm x,σ Ey,σ nm e jkn x x jkm y y (4.75) Pollaplasi zontac thn Ex. (4.75) eswterik me ton suzug enìc tuqaðou rujmoô Floquet (n,m,σ ) kai oloklhr ntac se ìlo to noigma thc op c, se sundiasmì me thn Ex. (4.74), ìpwc èqei perigrafeð kai prohgoumènwc, ta ìria sta oloklhr mata metaxô twn rujm n Floquet mporoôn na epektajoôn se ìl o to monadiaðo kelð kai qrhsimopoi ntac plèon tic idiìthtec pou anafèrjhkan sthn enìthta èqoume, (A s δ sσ + A p δ pσ ) δ,n m + ρn m σ = (C α + D α ) α wy 2 wy 2 wx 2 wx 2 ( ) E x,α E n m x,σ + E y,αe n m y,σ e +jkn x x+jkm y y dxdy (4.76) H parap nw exðswsh isqôei gia k je ènan apì touc rujmoôc Floquet, dhlad gia k je n,m,σ, ra to parap nw isodônameð me ena sônolo exis sewn. Sthn Ex. (4.76) ta δ sσ,δ pσ,δ,n m eðnai to dèlta tou Kronecker kai paðrnei tim Ðsh me th mon da mìno gia σ = s, σ = p, n = m = antðstoiqa. Dhlad faðnetai ìti o pr to ìroc emfanðzetai mìno ìtan h exiswsh prokôptei apì to eswterikì ginìmeno tou ìrou prìsptwshc thc antðstoiqhc pìlwshc, en antðjeta se k je llh perðptwsh rujmoô autìc o ìroc mhdenðzetai. OrÐzontac parak tw, S nm ασ = wy 2 wy 2 wx 2 wx 2 ( Ex,α E nm x,σ + E ) y,αey,σ nm e +jkx nx+jkm y y dxdy (4.77) 53

60 Kef laio 4. Mèjodoc Prosarmog c Rujm n h Ex.(4.76) gr fetai pio komy, (A s δ sσ + A p δ pσ ) δ,n m + ρnm σ = α S n m ασ (C α + D α ) (4.78) Apì thn Ex. (4.73b ) èqoume, ( Ys A s [ ] E y,s Ex,s α + Yp A p [ ] Ey,α Y α (C E α + D α ) x,α [ ]) E y,p Ex,p e jk xx jk yy + nmσ Yσ nm ρ nm σ [ ] E nm y,σ Ex,σ nm e jkn xx jky my = (4.79) Pollaplasi zoume thn Ex. (4.79) eswterik me to suzug k je rujmoô entìc twn op n ( β ), oloklhr noume se ìlh thn èktash tou anoðgmatoc twn op n kai qrhsimopoi ntac tic idiìthtec pou anafèrjhkan sthn par grafo kai ton orismì pou tèjhke sthn Ex. (4.77) èqoume, A s Ys S sβ + A pyp S pβ n,m,σ ρ nm σ Y σ nm Sσβ nm = Y β (C β D β ) (4.8) kai h parap nw exðswsh isqôei gia k je β dhlad gia k je rujmì entìc thc op c. AntÐstoiqa, efarmìzontac akrib c thn Ðdia diadikasða gia tic exis seic sunèqeiac tou hlektrikoô kai magnhtikoô pedðou sthn epif neia z = d, katal goume stic parak tw exis seic, t n m σ m e jkn z d = α S n m σ α ( Cα e jβz,αd + D α e +jβz,αd) n, m, σ (4.81) n,m,σ t nm σ Y nm σ e jknm z ( d = Y β S nm Cβ e jβz,βd D β e ) +jβ z,βd β TE, TM rujmoð σβ (4.82) dhlad gia k je rujmì entìc twn op n. Stic parap nw exis seic eis goume tic allagèc metablht n, E α = C α + D α, E α = C αe jβz,αd + D α e jβz,αd. (4.83a ) (4.83b ) Qrhsimopoi ntac tic nèec metablhtèc kai met apì pr xeic sticex.(4.78),(4.8), (4.81), (4.82) katal goume sto parak tw sôsthma exis sewn wc proc tic nèec metablhtèc E α,e α : (Σ α G αα ) E α β α G αβ E β Σ α E α = I α (4.84a ) Σ α E α +(Σ α G αα ) E α β α G αβ E β = (4.84b ) 54

61 Kef laio 4. Mèjodoc Prosarmog c Rujm n K je mða apì tic parap nw exis seic se mia tim tou deðkth α en oi posìthtec pou emfanðzontai orðzontai akoloôjwc: Σ α = Σ α = jy α tan β z,α d jy α sin β z,α d (4.85) (4.86) G αβ = nmσ Y nm σ Sσα nm Snm σβ (4.87) I α = 2 ( A s Ys S sα + A py nm p kai to G αα prokôptei apì thn Ex. (4.87) giaβ = α, Spα ) (4.88) G αα = nmσ Yσ nm Sσα nm 2 (4.89) Oi Ex. (4.84) sqhmatðzoun èna sôsthma exis sewn, ìpwc anafèrjhke parap nw, kaj c gia k je α, dhlad gia k je rujmì entìc thc op c èqoume dôo exis seic pou sundèoun ta E α, E α. Gr fontac autèc tic dôo exis seic gia k je ènan apì touc rujmoôc entìc twn op n, èqoume èna sôsthma pou eðnai sunepèc wc proc thn perigraf kai epilôsimo. An anaptôxoume to sôsthma twn exis sewn se morf pin kwn, prokeimènou na èqoume kalôterh eikìna thc summetrðac pou parousi zei, Σ 1 G 1,1 G 1,2 G 1,A Σ 1 G 2,1 Σ 2 G 2,2 G 2,A Σ G A,1 G A,2 Σ A G A,A Σ A Σ 1 Σ 1 G 1,1 G 1,2 G 1,A Σ 2 G 2,1 Σ 2 G 2,2 G 2,A Σ A G A,1 G A,2 Σ A G A,A E 1 E 2... E A E 1 E 2... E A = I 1 I 2... I A... AfoÔ lujeð to parap nw sôsthma, oi timèc twn ρ nm σ Ex. (4.78),(4.81), ρ nm σ = (A s δ s + A p δ p ) δ + α kai t nm σ prokôptoun apì tic S nm ασ E α (4.9) t nm σ = e jknm z d α S nm ασ E α (4.91) Prèpei na shmei soume ìmwc ìti se ìlec tic parap nw exis seic, prokeimènou autèc na perigrafoun th di taxh pl rwc, upojèsame ìti n, m =,...,+ kai ìti to α diatrèqei ìlouc touc pijanoôc rujmoôc entìc thc op c. Autì ìmwc eðnai praktik adônato na gðnei ìtan ja proqwr soume sthn arijmhtik epðlush tou parap nw sust matoc. 'Etsi eðnai anagkaðo na perioristoôn ta parap nw ajroðsmata se peperasmèno arijmì ìrwn all prèpei na tonisteð ìti to pl rec sôsthma twn peirwn ìrwn apoteleð thn orj perigraf tou arqikoô probl matoc. Sqetik me ton arijmì twn ìrwn pou prèpei na epilegeð ste h lôsh na sugklðnei sthn pragmatik, arkoôn mìno 3 5 ìroi rujm n Floquet kat x kai y prokeimènou na sugklðnoun ta ajroðsmata wc proc n, m, σ 55

62 Kef laio 4. Mèjodoc Prosarmog c Rujm n en gia touc rujmoôc entìc twn op n, o arijmìc twn ìrwn exart tai apì tic diast seic thc op c. An oi diast seic eðnai arket mikrìterec tou m kouc kômatoc, me apotèlesma ìloi oi rujmoð na eðnai aposbenômmenoi, arkeð Ðswc kai mìno o rujmìc qamhlìterhc t - xhc. An oi diast seic thc op c eðnai megalôterec, prèpei na prosarmìsoume ton arijmì twn ìrwn an loga. Prèpei na shmeiwjeð ìti, h par leiyh twn an terwn rujm n Floquet de ja prèpei na dhmiourgeð prìblhma sthn sôgklish thc lôshc gia ton akìloujo lìgo. 'Opwc perigr fhke sthn enìthta 4.1.2, to kumatikì di nusma kat x, y kai z enìc rujmoô dðnetai apì tic Ex. (4.6), kx n = k x + 2πn L x = k y + 2πm k m y k nm z = L y k 2 sup (k n x) 2 (k m y ) 2 (4.92a ) (4.92b ) (4.92g ) gia n, m =,..., ìpou parathroôme ìti gia touc ìrouc megalôterhc t xhc kai gia tic peript seic pou to L x,l y >λ, dhlad gia tic diat xeic pou ja mac apasqol soun, ta kx n,km y gðnontai arket meg la me apotèlesma to knm z na eðnai fantastikìc arijmìc. Praktik autì pou sumbaðnei se autèc tic peript seic eðnai ìti, oi ìroi Floquet an terhc t xhc eðnai ìroi aposbenômmenoi kai ra de fèroun mèroc thc diadidìmenhc isqôoc. 'Etsi gðnetai antilhptì, pwc h apaloif aut n twn ìrwn den ephre zei shmantik th sôgklish thc upologistik c lôshc me thn pragmatik. K ti apolôtwc an logo sumbaðnei kai me touc rujmoôc twn op n, kaj c ìso megalôterhc t xhc eðnai o rujmìc tìso pio aposbenômmenoc eðnai, me apotèlsma na fèrei mikrìtero kl sma thc isqôoc kai h par leiy tou na ephre zei ligìtero. Gia thn epðlush tou parap nw sust matoc exis sewn, ulopoi jhke me th bo jeia tou Matlab èna prìgramma epðlushc gia th genik perðptwsh di taxhc met llou me orjogwnikèc opèc. Ta aparaðthta dedomèna eisìdou tou progr mmatoc, eðnai ìlec oi paramètroi (diast seic gewmetrðac, suqnìthta, dieôjunsh prìsptwshc, analogða pol sewn) kai to apotèlesma pou prokôptei eðnai oi suntelestèc an klashc ρ nm σ kai met doshc t nm σ ìlwn twn ìrwn Floquet pou el fjhsan prokeimènou na lujeð to sôsthma kai gia ta dôo eðdh pol sewn. EpÐshc, prokeimènou na epilôsoume gia prospðpton kôma dedomènhc pìlwshc (mìno s mìno p), arkeð na tejeð o suntelest c A s,a p, gia thn s p pìlwsh, antðstoiqa, Ðsoc me th mon da kai o deôteroc Ðsoc me to mhden. O- poiosd pote lloc sundiasmìc dhmiourgeð mia kajorismènh pìlwsh. EpÐshc mèsw twn suntelest n aut n, pèra apì grammik pìlwsh, mporoôme na eis goume kai elleiptik, apl epilègontac kat llhlouc migadikoôc arijmoôc. Genik, parathroôme ìti h parap nw lôsh mac af nei meg lo eôroc epilog n wc proc to poia di taxh ja prosomoi soume kai ja qrhsimopoi jei stic prosomoi seic tou epìmenou kefalaðou. 56

63 Kef laio 5 Yeudo SPP rujmoð kai exairetik optik met dosh 5.1 Apì tic optikèc suqnìthtec stic mikrokumatikèc, apì touc SPP stouc yeudo SPP rujmoôc 'Opwc analôjhke kai sto kef laio 2, ta surface plasmon polaritons mporoôn na sugkentr soun to hlektromagnhtikì pedðo sth diepif neia metaxô met llou kai dihlektrikoô, se èktash polô mikrìterh tou m kouc kômatoc. Aut h uyhl sugkèntrwsh tou pedðou emfanðzetai mìno ìtan h suqnìthta tou hlektromagnhtikoô kômatoc brðsketai kont sth suqnìthta pl smatoc tou met llou, kaj c se aut thn perioq to pragmatikì mèroc thc dihlektrik c stajer c tou met llou gðnetai arnhtikì, epitrèpontac thn emf nish twn SPP rujm n. 'Etsi, oi shmantikìterec efarmogèc pou mporoôn na prokôyoun apì thn ekmet lleush twn SPP rujm n, ìpwc kumatod ghsh se perioqèc mikrìterec tou m kouc kômatoc kai optik anðqneush meg lhc akrðbeiac kai euaisjhsðac, periorðzontai sthn optik sthn kontin upèrujrh perioq tou f smatoc. Parathr ntac thn kampôlh diaspor c twn SPP rujm n, blèpoume ìti stic qamhlèc suqnìthtec, h stajer di doshc plhsi zei th stajer di doshc tou hlektromagnhtikoô kômatoc sto dihlektrikì, me apotèlesma th mikr sugkèntrwsh tou rujmoô sth diepif neia. Se autèc tic suqnìthtec, o rujmìc de dieisdôei sqedìn kajìlou sto mètallo, lìgw thc tèleiac agwgimìthtac pou parousi zoun ta mètalla se suqnìthtec èwc kai merik THz, kai ekteðnetai gia poll m kh kômatoc mèsa sto dihlektrikì. Apì ta parap nw, katal goume sto sumpèrasma, ìti h metatrop enìc uyhl sugkentrwmènou rujmoô se èna kôma pou praktik diadðdetai sto dihlektrikì par llhla me thn metallik epif neia, ofeðletai sth meðwsh thc dieðsdushc tou rujmoô entìc tou met llou. 'Ara afoô mia sqetik uyhl dieðsdush tou rujmoô kai sto eswterikì tou met llou eðnai aparaðthth prokeimènou na uposthrðzontai oi SPP rujmoð, katal goume kai fusik } plèon sto sumpèrasma ìti stic qamhlìterec suqnìthtec, èwc kai merik THz ìpou to mètallo sumperifèretai wc tèleioc agwgìc, den emfanðzontai SPP rujmoð. Sta plaðsia aut, èqoume plèon mia èndeixh gia to pwc ja mporoôsame na prosomoi soume touc SPP rujmoôc se autèc tic suqnìthtec. H lôsh eðnai, teqnht na 57

64 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh Συντελεστής μετάδοσης (%) Μήκος κύματος (nm) Sq ma 5.1: Peiramatik tim tou suntelest met doshc isqôoc mhdenik c t xhc se sôsthma tetragwnik n op n gia thn perðptwsh k jethc prìsptwshc (perðodoc di taxhc L = 75nm, mèso noigma op n 28nm) se fôllo Ag (p qoc h = 32nm). Eswterikì: Optik eikìna di taxhc me qr sh hlektronikoô mikroskopðou. auxhjeð h dieðsdush tou pedðou entìc tou met llou kai ènac trìpoc gia na gðnei autì eðnai na epexergastoôme thn epif neia tou met llou ste aut na mhn eðnai leða. Pr gmati, èqei apodeiqjeð ìti tèleioi agwgoð mporoôn na uposthrðxoun epifaneiakoôc rujmoôc pou omoi zoun me touc rujmoôc SPP, ìpwc touc èqoun dh perigrafeð, arkeð h epif neia touc na uposteð k poiou eðdouc diamìrfwsh. Oi rujmoð pou emfanðzontai se aut thn perðptwsh apokaloôntai yeudo SPP rujmoð (spoof plasmons designer plasmons). Gia thn ulopoðhsh twn yeudo SPP rujm n prèpei me k poio trìpo na eggrafeð sthn epif neia tou met llou èna sqèdio (pattern). To sqèdio autì mporeð na ekteðnetai mìno sth mða kai stic dôo diast seic thc epif neiac tou met llou, dhmiourg ntac ètsi mia monodi stath disdi stath periodikìthta antðstoiqa. To str ma pou prokôptei plèon paôei na sumperifèretai wc tèleioc agwgìc se ìlh thn èktash tou (afoô plèon den eðnai tètoioc, to mètallo periodik diakìptetai apì opèc aul kia) kai apokt mða nèa sumperifor. H idèa twn diamorfwmènwn epifanei n metallik n fôllwn den proèkuye mìno wc an gkh gia th dhmiourgða yeudo SPP rujm n stic qamhlìterec suqnìthtec, all tan èna jèma pou apasqìlhse kai peiramatik, kaj c sundèjhke ex' arq c me to fainìmeno pou onom sthke exairetik optik met dosh (extraordinary optical transmission). Gia arket qrìnia jewr jhke ìti anoðgmata mikrìtera tou m kouc kômatoc, parousi zoun polô mikrì suntelest met doshc tou fwtìc diamèsou tou met llou. Sta tèlh thc dekaetðac tou '9, stic ergasðec twn Ebbesen, Ghaemi, Kim [3] apodeðqjhke ìti, an 58

65 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh anoiqjoôn opèc se leptì metallikì fôllo, eðnai dunatìn na parathrhjeð to fainìmeno thc exairetik c optik c met doshc se m kh kômatoc èwc kai dèka forèc megalôtera tou anoðgmatoc twn op n, ìpwc faðnetai sto sq ma 5.1. To fainìmeno autì sundèjhke me th diègersh SPP rujm n sthn epif neia tou met llou kai th sôzeuxh twn rujm n aut n me hlektromagnhtikoôc rujmoôc entìc twn op n. Jewrhtikèc ergasðec pou asqol jhkan me thn perigraf tou fainomènou apèdeixan thn emf nish exairetik c optik c met doshc akìma kai se diat xeic ìpou to mètallo antimetwpðzetai wc tèleioc agwgìc. Pèra apì to fainìmeno thc exairetik c optik c met doshc, h upost rixh rujm n pou omoi zoun me touc epifaneiakoôc rujmoôc SPP sthn epif neia tèleiwn agwg n diamorfwmènwn me tetragwnikèc opèc problèfjhke apì ton Pendry [17]. Sta plaðsia aut c thc ergasðac, apodeðqjhke ìti h isodônamh energ dihlektrik stajer tou di trhtou met llou parousi zei parìmoia sumperifor me aut enìc met llou pou perigr fetai apì to montèlo Drude. Par to gegonìc ìti ta fainìmena twn yeudo SPP rujm n kai thc exairetik c optik c met doshc faðnetai na sundèontai mesa, sto parìn kef laio de ja asqolhjoôme tìso me th sôndesh twn dôo aut n fainomènwn, ìso me kaj' èna apì aut xeqwrist, kaj c kai ta dôo anex rthta mporoôn na broun shmantikèc efarmogèc. Oi prosomoi seic pou akoloujoôn gðnontai se di taxeic an klashc kai met doshc, mon c kai dipl c periodikìthtac (aul kia egkopèc sthn perðptwsh thc mon c periodikìthtac kai opèc sth perðptwsh thc dipl c) kai tèloc parametrik wc proc ta gewmetrik qarakthristik. Sto parìn kef laio ja parousiastoôn k poiec prosomoi seic diat xewn, tìso monodi stathc, ìso kai disdi stathc periodikìthtac. Ja xekin soume me gewmetrðec pou sqetðzontai me to fainìmeno thc exairetik c optik c met doshc kai tèloc ja oloklhr soume me thn parousðash diagramm twn pou apodeiknôoun thn emf nish twn yeudo SPP sth diepif neia di trhtou met llou kai dihlektrikoô se suqnìthtec ìpou to mètallo mporeð na jewrhjeð wc tèleioc agwgìc. 5.2 Exairetik Optik Met dosh Metallikì fôllo me orjogwnik anoðgmata TM pìlwsh Sthn par grafo aut ja prosomoiwjeð h gewmetrða metallikoô fôllou me orjogwnik anoðgmata sth mða di stash, ìpwc faðnetai sto sq ma 5.2(a ) kai ja parousiastoôn ta diagr mmata tou suntelest met doshc mhdenik c t xhc sunart sei tou m kouc kômatoc tou prospðptontoc kômatoc. H sumperifor thc di taxhc ja melethjeð wc proc tic ex c paramètrouc: p qoc d tou metallikoô fôllou gia k jeth prìsptwsh, pl toc orjogwnik n anoigm twn w epðshc gia k jeth prìsptwsh, gwnða prìsptwshc θ inc gia stajerì p qoc met llou kai pl toc anoigm twn. 59

66 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh k inc k inc E I II III y w z xx x d inc - L/2 - w/2 w/2 - L/2 x Metal d L (αʹ) z (βʹ) Sq ma 5.2: (a) Trisdi stath apeikìnish thc gewmetrðac metallikoô fôllou me orjogwnik egkop kai (b) monadiaðo kelð thc gewmetrðac sto epðpedo xz. Sumperifor sunart sei tou p qouc tou met llou Sto sq ma 5.3, parousi zetai h sumperifor tou suntelest met doshc mhdenik c t xhc gia di taxh met llou me L=1.75mm, w =.3mm kai d apì.1 èwc.8mm. ParathroÔme ìti se ìlec tic peript seic epitugq netai suntelest c met doshc Ðsoc me th mon da. Autì pou prokôptei kurðwc apì to sq ma 5.3, eðnai ìti h jèsh tou suntonismoô metatopðzetai proc megalôtera m kh kômatoc (mikrìterec suqnìthtec) ìso aux netai to p qoc tou met llou. H emf nish twn suntonism n aut n sth sumperifor tou suntelest met doshc basðzetai sto ex c fainìmeno. To epðpedo kôma pou prospðptei sthn epif neia suzeôgnutai me ton jemeli dh rujmì entìc tou orjogwnikoô anoðgmatoc. To orjogwnikì noigma sumperifèretai wc mða koilìthta Fabry Perot kaj c to kôma kineðtai sto eswterikì tou kai upìkeitai se allep llhlec anakl seic stic epif neiec z =kai z = d lìgw thc diafor c thc kumatik c antðstashc tou rujmoô sto eswterikì twn op n me thn kumatik antðstash stic perioqèc I kai III. Gia mia sugkekrimènh suqnìthta h allhlepðdrash metaxô twn anakl sewn odhgeð sthn emf nish suntonismoô opìte parathroôme kai to mègisto tou suntelest met doshc. H sumperifor tou suntelest met doshc sunart sei tou p qouc tou metallikoô fôllou eðnai anamenìmenh me b sh thn parap nw an lush, kaj c ìso aux nei to p qoc tou met llou, isodônama aux nei to m koc tou suntonist Fabry Perot kai oi suntonismoð parathroôntai se megalôtera m kh kômatoc. To di gramma 5.3 epibebai netai kai apì th bibliografða sthn ergasða twn Martin Moreno, Garcia Vidal [18]. Sumperifor sunart sei tou anoðgmatoc twn aulaki n Diathr ntac stajerì to p qoc tou met llou, metab lloume to noigma twn aulaki n, apeikonðzontac antðstoiqa ton suntelest met doshc mhdenik c t xhc sunart sei tou m kouc kômatoc. Oi diast seic eðnai se aut thn perðptwsh L =1.75mm, d =.2mm kai w apì.1 èwc 1mm. Apì to sq ma 5.4 parathroôme ìti h suqnìthta sthn opoða emfanðzetai o suntonismìc metab lletai el qista kaj c aux netai to pl toc twn anoigm twn (egkop n), gegonìc pou epibebai nei thn upìjesh ìti o suntonismìc ofeðletai sthn emf nish fainomènou parìmoiou me to fainìmeno Fabry Perot kai ra h jèsh twn suntonism n 6

67 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh T (m) x 1-6 d=.1 μm d=.2 μm d=.4 μm d=.6 μm d=.8μm Sq ma 5.3: F smatikì di gramma tou suntelest met doshc mhdenik c t xhc T gia diaforetik p qh met llou. Par metroi gewmetrðac: L =1.75mm, w =.3mm kai d apì.1 èwc.8mm. Ta diagr mmata parousi zontai auxhmèna kata 1 k je for gia kalôterh apeikìnish. ParathroÔme ìti se k je perðptwsh p qouc met llou o suntelest c met doshc parousi zei sumperifor suntonismoô, h suqnìthta tou opoðou metatopðzetai proc megalôtera m kh kômatoc kaj c aux netai to p qoc d. Dihlektrikì upostr matoc, uperstr matoc eðnai o aèrac (n =1). sqetðzetai me to p qoc tou met llou. Auto pou faðnetai na ephre zetai shmantik se aut thn perðptwsh, eðnai to eôroc twn suntonism n kai h mèsh st jmh tou suntelest met doshc pou aux netai ìso aux nei to pl toc twn egkop n. Wc mèsh st jmh anafèretai h tim tou suntelest met doshc mhdenik c t xhc makri apì thn perioq tou suntonismoô. 'Opwc exhg jhke kai parap nw, o suntonismìc ofeðletai sthn emf nish tou fainomènou Fabry Perot an mesa stic dôo epif neiec epaf c tou metallikoô fôllou me to upèrstrwma kai to upìstrwma. 'Oso megal nei to pl toc twn anoigm twn, pèra apì ton suntonismì o opoðoc emfanizetai se sugkekrimènh suqnìthta, èna pl joc llwn suqnot twn mporeð na suzeuqjeð me rujmoôc sthn perioq met doshc, me mikrìterh bèbaia apodotikìthta, lìgw tou auxhmènou anoðgmatoc twn aulaki n. 'Etsi kai h mèsh st jmh tou suntelest met doshc aux nei kaj c megal nei to noigma. Sumperifor sunart sei thc gwnðac prìsptwshc H allag thc gwnðac prìsptwshc pèra apì tic sthn perðptwsh thc k jeth prìsptwshc, ephre zei th sunist sa tou kumatikoô dianôsmatoc tou prospðptontoc fwtìc kat th dieôjunsh x afoô isqôei, k x = k sin (θ inc ). (5.1) 61

68 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh T w=.1 w=.2 w=.4 w=.6 w=.8 w=1 μm μm μm μm μm μm (m) Sq ma 5.4: Fasmatikì di gramma tou suntelest met doshc mhdenik c t xhc sunart sei tou anoðgmatoc twn aulaki n. Par metroi gewmetrðac: L = 1.75mm, d =.2mm kai w =.1èwc 1mm. K je epìmeno di gramma x 1-6 èqei metatopisteð kat 1 proc ta p nw gia kalôterh optik apeikìnish. ParathroÔme ìti, h metabol tou pl touc twn anoðgmatoc den ephre zei shmantik th jèsh twn suntonism n all odhgeð se aôxhsh tou eôrouc. Dihlektrikì upostr matoc, uperstr matoc eðnai o aèrac (n =1). Me dedomèno ìti h lôsh tou probl matoc èqei gðnei gia th genik perðptwsh, anex rthta apì thn tim tou kx, mporoôme me to Ðdio prìgramma pou gr fthke kai epilôei thn perðptwsh thc k jethc prìsptwshc, na epilôsoume kai aut th genik perðptwsh prìsptwshc. H prosomoðwsh pou ja akolouj sei ja gðnei gia mia seir gwni n prìsptwshc apì èwc kai 5, eôroc pou eðnai entìc twn anamenìmenwn fusik n orðwn gia k poio prospðpton kôma. Ja apeikonðsoume ton suntelest met doshc se èna di - gramma epif neiac, ìpou ston orizìntio xona topojeteðtai h gwnða prìsptwshc kai ston katakìrufo to m koc kômatoc. Oi diast seic thc di taxhc eðnai L =1.75mm, d =.2mm, w =.3mm kai to m koc kômatoc, λ, kineðtai an mesa sta 1.4kai 3mm. Oi suntonismoð entopðzontai sto di gramma, gôrw apì tic perioqèc tou skoôrou mplè qr matoc kaj c, ìpwc èqoume parathr sei wc t ra, k je suntonismìc sunodeôetai apì mhdenismoôc tou suntelest met doshc se lðgo mikrìtero kai lðgo megalôtero m koc kômatoc kai eðnai arket oxôc me apotèlesma na mh emfanðzetai arket kajar. ParathroÔme sto sq ma tic suneq c grammèc pou sqhmatðzoun ta mègista tou suntelest met doshc kai ètsi apoktoôme mða eikìna gia th metabol tou suntelest met doshc sunart sei thc gwnðac prìsptwshc tou epðpedou kômatoc. 62

69 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh Sq ma 5.5: Q rthc tou suntelest met doshc sunart sei thc gwnðac prìsptwshc (orizìntioc xonac) kai tou m kouc kômatoc (katakìrufoc xonac). Ta shmeða pou antistoiqoôn sta mègista tou suntelest met doshc brðskontai an mesa stouc mhdenismoôc (skoôro mplè qr ma) kai diakrðnontai kajar sthn perioq aut Metallik epif neia me orjogwnikèc opèc H an lush pou prohg jhke gia thn perðptwsh twn orjogwnik n monodi statwn anoigm twn se metallikì fôllo, dustuq c de mporeð na bohj sei sthn katanìhsh twn idiot twn met doshc thc di taxhc orjogwnik n op n se metallikì fôllo, kaj c mia orjogwnik op den uposthrðzei TEM rujmoôc se antðjesh me ta orjogwnik anoðgmata mðac di stashc. Gia to l ìgo autì, efarmìsthke h an lush thc paragr fou 4.3, h opoða ìmwc eðnai saf c pio apaithtik apì thn poyh thc apaitoômenhc upologistik c isqôoc, kaj c to sôsthma parousi zei periodikìthta se dôo diast seic. Sthn ergasða twn Popov et al. [19] parousi sthke ènac upologismìc me th qr sh thc mejìdou epèktashc rujm n, lamb nontac up' ìyin thn dihlektrik stajer tou met llou sunart sei thc suqnìthtac ε(ω). Oi upologismoð autoð èdeixan thn Ôparxh megðstwn ston suntelest met doshc thc di taxhc. Ta mègista aut apodìjhkan jewrhtik stouc hlektromagnhtikoôc rujmoôc pou diadðdontai sto eswterikì twn op n kai parathr jhkan ìtan gia to ε(ω) jewr jhkan timèc pragmatik n met llwn, en ta mègista den emfanðsthkan ìtan to mètallo jewr jhke wc tèleioc agwgìc. Sthn paroôsa par grafo, me th bo jeia thc an lushc pou prohg jhke gia to sôsthma metallikoô fôllou me orjogwnikèc opèc, ja apodeðxoume thn emf nish megðstwn tou suntelest met doshc mhdenik c t xhc, akìma kai ìtan to mètallo antimetwpðzetai wc tèleioc agwgìc. H prosomoðwsh ja gðnei se di taxh metallikoô fôllou me L x = L y = 75nm, w x = w y = 284nm kai d = 2nm. Ja epilèxoume N=1, M=, A=1 dhlad 3 ruj- 63

70 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh L x Ly wy wx x k inc d I y II z III Sq ma 5.6: MonadiaÐo kelð gewmetrðac met llou me orjogwnikèc opèc.to f c prospðptei apì thn perioq I, h perioq II antistoiqeð sto mètallo kai tic periodikèc opèc, en h perioq III eðnai h perioq thc met doshc. moôc Floquet kat th dieôjunsh x, mìno ton mhdenikì rujmì kat th dieôjunsh y kai mìno touc jemeli deic TE kai TM rujmoôc entìc twn op n. To prospðpton kôma eðnai polwmèno sth dieôjunsh p. H epilog aut ègine prokeimènou ta apotelèsmata pou ja prokôyoun na mporoôn na sugkrijoôn me ta antðstoiqa apotelèsmata pou parousi zontai sthn ergasða twn Martin Moreno, Garcia Vidal [18]. Sthn anaferìmenh ergasða [18], h emf nish tou fainomènou thc exairetik c optik c met doshc exhgeðtai me b h ton akìloujo sullogismì. To prospðpton hlektromagnhtikì kôma suzeôgnutai me mh sugkentrwmèno (aposbennômeno, leaky) yeudo SPP rujmì se k je mða apì tic diepif neiec met llou aèra. Oi rujmoð autoð allhlepidroôn metaxô touc mèsw twn hlektromagnhtik n rujm n pou diadðdontai sto eswterikì twn op n, dhmiourg ntac summetrikèc kai antisummetrikèc katast seic tou kômatoc. H emf nish tou fainomènou thc exairetik c optik c met doshc sundèetai me thn krðsimh} sôzeuxh metaxô twn rujm n aut n. Sto sq ma 5.7 parathroôme emfan c ta dôo mègista tou suntelest met doshc pou emfanðzontai, to pr to gia m koc kômatoc λ =1.5L x kai to deôtero gia λ = 1.26L x. To di gramma autì èqei sugkrijeð kai sumfwneð exairetik me to antðstoiqo pou emfanðzetai sthn ergasða twn Moreno,Vidal [18]. Sto sq ma 5.8, parousi zetai mia leptomèreia tou diagr mmatoc 5.7 apì ta 74nm èwc ta 78nm gia thn Ðdia gewmetrða, epilôontac ìmwc to sôsthma twn exis sewn me perissìterouc ìrouc, tìso rujm n Floquet ìso kai rujm n entìc twn op n. ParathroÔme ìti kaj c aux noume ton arijmì twn ìrwn, dhlad belti noume th sôgklish thc mejìdou, oi suntonismoð metatopðzontai proc mikrìtera m kh kômatoc. H morf thc kampôlhc den alloi netai, apl apoktoôme kalôterh eikìna gia thn pragmatik jèsh twn suntonism n tou suntelest met doshc pou parousi zei h gewmetrða. EÐnai shmantikì na anafèroume ìti h aôxhsh twn ìrwn N, M, A aux nei dramatik thn apaitoômenh upologistik isqô kai gi' autì to lìgo sth sunèqeia, pou ja qreiasteð na gðnoun analôseic wc proc to m koc kômatoc kai wc proc ta qarakthristik tou kumatikoô dianôsmatoc thc prìsptwshc, ja epilegeð o el qistoc arijmìc ìrwn gia ta N, M, A ste h lôsh na prokôyei se logikì qrìno upologismoô kai me ikanopoihtik akrðbeia. 64

71 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh T N=1, M=, A= x 1-7 (m) Sq ma 5.7: Di gramma tou suntelest met doshc mhdenik c t xhc sunart sei tou m kouc kômatoc. ParathroÔme ta dôo mègista pou emfanðzontai ston suntelest met doshc kai sundèontai me to fainìmeno thc exairetik c optik c met doshc gia λ = 1.5L x kai λ = 1.26L x. Par metroi gewmetrðac L x = L y = 75nm, w x = w y = 284nm kai d = 2nm T N=M=4, A=2 N=1, M=,A=1 N=M=2, A= x 1-7 (m) Sq ma 5.8: Di gramma tou suntelest met doshc mhdenik c t xhc T gia th di taxh pou anafèretai kai to sq ma 5.7, esti zontac sthn perioq twn mhk n kômatoc pou emfanðzetai o suntonismìc. Parousi zontai apotelèsmata me th qr sh perissìterwn ìrwn epðpedwn kum twn kai rujm n entìc twn op n. H qr sh perissìterwn ìrwn faðnetai na metatopðzei tic jèseic twn megðstwn tou suntelest met doshc proc mikrìtera m kh kômatoc. 65

72 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh 4.5 T d=.1l x d=.3l x d=.5l x d=.7l x d=.9l x (m) x 1-7 Sq ma 5.9: Suntelest c met doshc mhdenik c t xhc T, gia diaforetik p qh metallikoô fôllou me tetragwnikèc opèc. Par metroi gewmetrðac L x = L y = 75nm, w x = w y = 284nm kai d apì.1l x èwc.9l x. Ta diagr mmata metatopðzontai kat 1 proc ta p nw gia kalôterh apeikìnish. ParathroÔme ìti, kaj c aux nei to p qoc tou met llou oi suntonismoð plhsi zoun metaxô touc kai gia meg lo p qoc, telik q nontai. Melèth sunart sei tou p qouc tou met llou Sthn upoenìthta aut akoloujeð h an lush tou diagr mmatoc tou suntelest met doshc sunart sei tou p qouc tou metallikoô fôllou, an loga me thn perðptwsh tou metallikoô fôllou me orjogwnikèc egkopèc. Wc anafor, ja jewrhjeð h di taxh pou melet jhke parap nw sthn opoða èqoume, L x = L y = 75nm, w x = w y = 284nm kai d = 2nm. Ja prosomoi soume th di taxh sta Ðdia m kh kômatoc, diathr ntac stajerèc tic upìloipec diast seic kai ja metab lloume to p qoc apì to.1l x èwc kai.9l x. Ta diagr mmata pou paratðjontai sto sq ma 5.9 metatìpizontai me th seir kat 1 proc ta p nw gia kalôterh apeikìnish. Gia thn epðlush tou probl matoc èqoume epilèxei N=1, M=, A=1. Apì to di gramma 5.9 parathroôme ìti kaj c aux nei to p qoc tou metallikoô fôllou, oi suntonismoð tou suntelest met doshc plhsi zoun metaxô touc kai gia thn tim d =.5L x ekfulðzontai se ènan mìno suntonismì, en ìso aux noume perissìtero to p qoc tou metallikoô fôllou, parathroôme ìti to mègisto tou suntelest met doshc mei netai. Autì eðnai anamenìmeno sumpèrasma, kaj c ìso aux nei to p qoc tou met llou mei netai h allhlepðdrash metaxô twn epifaneiak n rujm n sth diepif neia met llou aèra kai ra mei netai h dunatìthta sôzeuxhc touc, gia thn emf nish tou fainomènou thc exairetik c optik c met doshc. Autì ofeðletai kai sto gegonìc ìti, gia tic sugkekrimènec suqnìthtec kai to sugkekrimèno noigma twn op n, ìloi oi ruj- 66

73 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh moð entìc twn op n eðnai aposbennômenoi, opìte to m koc di doshc touc eðnai sqetik mikrì, me apotèlesma gia megalôtero p qoc met llou na mei netai to mèroc thc isqôoc pou ft nei telik sthn perioq III kai mporeð na suzeuqjeð me rujmoôc se ekeðnh thn perioq. ParathroÔme ìti gia d =.9L x to mègisto tou suntelest met doshc sqedìn q netai. H diafor se sqèsh me thn perðptwsh twn orjogwnik n egkop n eðnai ìti ekeðnec oi gewmetrðec emf nizan suntelest met doshc Ðso me th mon da akìma kai gia meg la p qh tou metallikoô fôllou, en sthn gewmetrða tetragwnik n op n den parathreðtai k ti an logo. H diaforopoðhsh aut prokôptei apì to gegonìc ìti h gewmetrða tetragwnik n/orjogwnik n op n den uposthrðzei TEM rujmoôc (oi opoðoi den èqoun suqnìthta apokop c) antðjeta me thn perðptwsh twn egkop n, ìpou kai gia polô meg lo p qoc met llou h prospðptousa enèrgeia mporeð na suzeuqjeð me ton basikì TEM rujmì o opoðoc den eðnai aposbennômenoc kai telik ft nei wc thn perioq thc met doshc. Melèth sunart sei tou anoðgmatoc twn op n Sta sq mata 5.1, 5.11 parousi zetai h metabol tou suntelest met doshc mhdenik c t xhc sunart sei tou anoðgmatoc twn op n. Se aut thn perðptwsh thc prosomoðwshc arqðzoume p li me di taxh anafor c th di taxh pou orðsthke sthn arq gia thn opoða isqôei, L x = L y = 75nm, d = 2nm kai to pl toc thc op c, w x,w y ja metab lletai apì.2l x èwc.8l x. H an lush aut c thc perðptwshc ja gðnei se dôo qwrist diagr mmata gialìgouc pou ja fanoôn parak tw. ParathroÔme sta sq mata 5.1, 5.11 ìti h uxhsh tou megèjouc twn op n odhgeð sthn apom krunsh twn dôo suntonism n. Gia to lìgo autì epilèqjhke gia mègejoc o- p c megalôtero apì.6l x na apeikonðsoume ton suntelest met doshc se diaforetikèc klðmakec mhk n kômatoc. Gia megalôtera anoigmata op n epilèxame na epekteðnoume thn an lush mèqri to 1mm. 'Opwc kai sthn perðptwsh twn monodi statwn orjogwnik n anoigm twn, h aôxhsh tou megèjouc twn op n epitrèpei thn eukolìterh sôzeuxh tou prospðptontoc kômatoc me rujmoôc met doshc sthn perioq III aux nontac ètsi, tìso to eôroc twn suntonism n, ìso kai th st jmh sthn opoða ft nei o suntelest c met doshc mhdenik c t xhc makri apì ta shmeða suntonismoô. Akìma kai se autèc tic peript seic ìmwc emfanðzontai shmeða mhdenismoô tou suntelest met doshc lìgw pijan c k poiou fainomènou an logou tou fainomènou Fabry Perot me katastreptikì ìmwc qarakt ra se aut thn perðptwsh. Melèth sunart sei thc gwnðac prìsptwshc θ inc Sthn ergasða tou Hibbins [2] ègine mða peiramatik melèth thc allhlepðdrashc epðpedou kômatoc kai gewmetrðac met llou me tetragwnikèc opèc. H peiramatik melèth ègine stic suqnìthtec 18 èwc 26GHz kai melet jhke h sumperifor tou suntelest met doshc sunart sei thc gwnðac zenðj, θ inc, kai thc suqnìthtac. Gia k je mða apì tic suqnìthtec melèthc, up rqe metabol thc gwnðac θ inc apì tic èwc kai 5. Me b sh tic peiramatikèc autèc metr seic, sqhmatðsthke ènac q rthc tou suntelest met doshc mhdenik c t xhc sunart sei thc suqnìthtac tou prospðptontoc kômatoc kai tou mètrou tou kumatikoô dianôsmatoc kat th dieôjunsh x. UpenjumÐzoume ìti to mètro tou kumatikoô dianôsmatoc kat th dieôjunsh x, k x, sundèetai me ton kumatikì arijmì 67

74 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh 4 T w=.1l x w=.2l x w=.3l x w=.4lx (m) x 1-7 Sq ma 5.1: Suntelest c met doshc mhdenik c t xhc T gia diaforetik anoðgmata twn op n. Par metroi gewmetrðac L x = L y = 75nm,d = 2nm kai w x = w y apì.1l x èwc.4l x T w =.6L x w =.8L x x 1 (m) -7 Sq ma 5.11: Suntelest c met doshc mhdenik c t xhc T gia diaforetik anoðgmata twn op n. Par metroi gewmetrðac L x = L y = 75nm,d = 2nm kai w x = w y apì.6l x èwc.8l x. H klðmaka twn mhk n kômatoc èqei all xei kai ekteðnetai mèqri to 1mm. ParathroÔme thn apom krunsh twn suntonism n kai thn aôxhsh thc st jmhc tou suntelest met doshc makri apì touc suntonismoôc, ìso aux nei to noigma twn op n. 68

75 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh sthn perioq thc prìsptwshc, k, mèsw thc sqèshc, k x = k cos φ inc sin θ inc (5.2) ìpou φ inc h azimoujiak gwnða metrhmènh wc proc ton xona x kai θ inc h gwnða zenðj metrhmènh wc proc thn katakìrufo sthn epif neia prìsptwshc. KalÔterh eikìna thc gewmetrðac parèqetai anatrèqontac sto sq ma 5.6. Se aut thn par grafo ja parousiasteð to di gramma pou prokôptei upologistik me th qr sh tou progr mmatoc pou ulopoieð th mèjodo prosarmog c rujm n, gia th di taxh pou metr jhke peiramatik sto anaferìmeno rjro. Oi par metroi thc gewmetrðac eðnai, L x = L y =9.53mm, w x = w y =6.96mm, d =15mm h suqnìthta kumaðnetai an mesa sta 18 26GHz kai θ inc stic èwc 5. Sto sq ma5.12 parousi zetai h pragmatik di taxh pou metr jhke peiramatik, en sto sq ma 5.13 parousi zontai ta apotelèsmata thc prosomoðwshc. ParathroÔme ìti sqhmatðzontai suneqeðc grammèc megðstwn tou suntelest met doshc mhdenik c t xhc (èntonec leukèc grammèc). H emf nish twn megðstwn sundèetai me thn emf nish aktinoboloômenwn yeudo SPP rujm n (leaky modes) stic dôo epif neiec met llou aèra kai th sôzeuxh metaxô touc. Ta apotelèsmata pou proèkuyan me thn prosomoðwsh sumfwnoôn polô ikanopoihtik me ta peiramatik apotelèsmata pou parousi zontai sthn anaferìmenh ergasða [2]. 69

76 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh d L x w x Sq ma 5.12: DokÐmio mètrhshc [2]. Ta gewmetrik qarakthristik èqoun tic timèc L x = L y =9.53mm, w x = w y =6.96mm, d =15mm, h suqnìthta kumaðnetai an mesa sta GHz kai θ inc apì èwc 5 kai oi peiramatikèc metr seic mporoôn na brejoôn sto anaferìmeno rjro tou Hibbins [2]. 2.6 x 11 T f (Hz) k x Sq ma 5.13: Apotèlesma prosomoi shc thc di taxhc pou emfanðzetai sto sq ma 5.12 me th qr sh thc mejìdou prosarmog c rujm n. ParathreÐtai h dipl gramm twn megðstwn tou suntelest met doshc, (kìkkinec grammèc). 7

77 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh 5.3 Suzeugmènoi yeudo SPP rujmoð Apì thn an lush pou prohg jhke sto kef laio 2, f nhke ìti oi SPP rujmoð emfanðzontai sth diepif neia dôo ulik n me antðjeto prìshmo tou pragmatikoô mèrouc thc dihlektrik c stajer c. EÐnai profanèc loipìn, ìti tètoioi rujmoð den mporoôn na uposthriqjoôn sth diepif neia metaxô enìc tèleiou agwgoô kai dihlektrikoô. O periorismìc autìc mporeð na xeperasteð an to mètallo diamorfwjeð periodik sth mða stic dôo diast seic meorjogwnik anoðgmata opèc, antðstoiqa, ìpwc èqoume dh anafèrei. H anaz thsh kai h epibebaðwsh thc Ôparxhc twn yeudo SPP rujm n, ja basisteð sthn exagwg tou diagr mmatoc diaspor c twn rujm n pou uposthrðzontai sth diepif neia metaxô diamorfwmènou tèleiou c agwgoô kai dihlektrikoô. UpenjumÐzoume ìti stouc klasikoôc SPP rujmoôc, h sqèsh diaspor c sundèei th suqnìthta me to kumatikì di nusma kat th dieôjunsh di doshc, dhlad th suqnìthta me th stajer di doshc. An loga, sthn perðptwsh twn diat xewn allhlepðdrashc tèleiou met llou dihlektrikoô me epðpeda kômata, o rujmìc pou ja sqhmatðzetai, ìntac apì th fôsh tou epifaneiakìc, ja diadðdetai kat th dieôjunsh x y. Prokeimènou na aplopoi soume k pwc thn an lush, tuqaða epilègetai ìti se k je di taxh pou ja melethjeð, ja isqôei ìti ky =kai ra h anaz thsh thc Ôparxhc tou rujmoô ja gðnei sth dieôjunsh x. Praktik dhlad, h sunistamènh tou kumatikoô dianôsmatoc tou prospðptontoc kômatoc kat th dieôjunsh x, k x, ja sumpðptei me th stajer di doshc tou rujmoô pou diereun tai. 'Etsi to di gramma diaspor c enìc yeudo SPP rujmoô ja apeikonðzetai se di gramma k x k k x ω. Apì thn an lush pou ègine sto kef laio 2, apodeðqjhke ìti prokeimènou oi SPP rujmoð na eðnai suzeugmènoi sthn epif neia epaf c twn dôo ulik n kai na aposbènontai ekjetik ìso apomakrunontai apì thn epif neia, sto egk rsio wc proc th di dosh epðpedo, prèpei h stajer di doshc, ra kai h kampôlh diaspor c, na brðsketai dexi apì thn gramm fwtìc tou dihlektrikoô. H idiìthta aut twn tupik n SPP rujm n, upodeiknôei thn perioq tou diagr mmatoc diaspor c thn opoða prèpei na anazhthjoôn oi yeudo-spp rujmoð. Sthn perðptwsh twn rujm n aut n, prokeimènou kai p li na eðnai suzeugmènoi kai ìqi aktinoboloômenoi rujmoð, prèpei h kampôlh diaspor c na brðsketai dexi thc gramm c fwtìc tou dihlektrikoô tou uperstr matoc. Sthn perioq dexi thc gramm c fwtìc tou dihlektrikoô isqôei ìti k x >k kai ra, k z = k 2 k2x = j kx 2 k2. (5.3) Apì thn Ex. (5.3) faðnetai ìti to prospðpton kôma sthn epif neia tou met llou, den eðnai diadidìmeno all sthn pragmatikìthta aposbennômeno. Autì shmaðnei ìti to prìblhma den eðnai kal orismèno apì fusik c pleur c kai ètsi opoiad pote perðergh} sumperifor tou suntelest an klashc met doshc den prèpei na antimetwpisteð a priori wc sf lma. H exagwg tou diagr mmatoc diaspor c gia thn perðptwsh twn yeudo SPP rujm n ja gðnei apo to q rth} 1 tou suntelest an klashc mhdenik c t xhc wc proc ta k x,k. H emf nish twn yeudo SPP rujm n sundèetai me apoklðseic stic timèc tou suntelest an klashc mhdenik c t xhc. 'Ara ta mègista ta el qista tou suntelest an klashc 1 Χάρτης εδώ νοείται ένα διάγραμμα επιφάνειας του μέτρου του συντελεστή ανάκλασης μηδενικής τάξης συναρτήσει των k x,k 71

78 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh pou ja sqhmatðzoun suneqeðc kampôlec stonq rthk x k ja upodhl noun thn emf nish yeudo SPP rujm n sta shmeða aut Yeudo SPP rujmoð sth diepif neia met llou me tetragwnikèc opèc peirou b jouc kai dihlektrikoô H an lush arqik, ja perioristeð sthn perðptwsh met llou me tetragwnikèc opèc polô meg lou b jouc (jewrhtik peirou) (w x = w y = w). Oi Pendry, Moreno, Vidal [21] prìteinan mða hmianalutik mèjodo gia th exagwg thc sqèshc diaspor c twn yeudo SPP rujm n. H mèjodoc aut sthrðzetai sth qr sh enìc el qistou montèlou me b sh th mèjodo prosarmog c rujm n, thn exagwg mèsw autoô thc analutik c èkfrashc tou suntelet an klashc mhdenik c t xhc kai sth sunèqeia thn exagwg thc sqèshc diaspor c apì ta shmeða apeirismoô (apìklishc) tou suntelest an klashc. To el qisto montèlo, pou apoteleð kai thn aploôsterh prðptwsh epðlushc, genik eis gei tic ex c aplopoi seic, To prospðpton kôma apoteleðtai mìno apì kôma p pìlwshc kai prospðptei p nta upì gwnða φ inc =(metrhmènh wc proc ton xona x). jewroôme ìti to anakl meno kôma, pou genik analôetai se mða seir epðpedwn kum twn kat to je rhma Floquet Bloch se aut thn perðptwsh apoteleðtai mìno apì ton ìro mhdenik c t xhc kai me pìlwsh ìmoia thc arqik c, sto eswterikì twn op n lamb netai upfloyin mìno o rujmìc me th mikrìterh suqnìthta apokop c, sth sugkekrimènh perðptwsh o TE 1 rujmìc, den emfanðzetai entìc twn op n ìroc di doshc proc ta arnhtik z, kaj c lìgw tou peirou b jouc twn op n, den up rqei an klash sthn epif neia epaf c twn perioq n II kai III, h efarmog thc sunj khc sunèqeiac periorðzetai mìno sthn epif neia z =, kaj c praktik h perioq III den emfanðzetai kajìlou. Epib llontac ìlec tic parap nw aplopoi seic stic exis seic thc enìthtac 4.3 è- qoume, E I = E inc + ρ E ref (5.4) ìpou, E inc = 1 [ ] 1 e jk x x jk z z (5.5) L x Sthn perioq entìc twn op n, E II = [ ] Exα C E α e jβαz (5.6) yα ìpou [ ] Exα = E yα 2 [ π w sin w ( y + w 2 )] (5.7) 72

79 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh An loga me thn enìthta 4.3 prokôptoun kai oi ekfr seic gia to mègejoc tou magnhtikoô pedðou. Efarmìzontac th sunj kh sunèqeiac sthn epif neia z =, èqoume Y p [ ] E xp Eyp [ ] E e jk xx + ρ xp Eyp [ ] E yp Exp e jk x x ρ Yp [ ] E yp Exp 2 [ π e jk x x = C α w sin w e jk x x = C α Y α ( y + w 2 2 w sin [ π w )] (5.8) ( y + w 2 )] (5.9) kai akolouj ntac thn Ðdia mejodologða me thn enìthta 4.3, pollaplasi zoume thn [ ] E ] xp Ex. (5.8)me e jk x x Eyα, en thn Ex. (5.9)me[ kai oloklhr noume sto noigma E yp thc tetragwnik c op c ( x w x /2, y w y /2). Sthn olokl rwsh thc Ex. (5.8), upenjumðzoume ìti sto olokl rwma metaxô twn rujm n Floquet, ta ìria olokl rwshc apì w/2...w/2 metab llontai se L/2...L/2 kaj c epib lloume to mhdenismì thc efaptomenik c sunist sac tou hlektrikoô pedðou stic perioqèc x >w x /2 kai y >w y /2. UpenjumÐzoume epðshc ìti to eswterikì ginìmeno twn ìrwn, tìso entìc twn op n, ìso kai sthn perioq thc an klashc, me ton eautì touc dðnei mon da. 'Etsi katal goume ìti, E xα 1+ρ = C α S (5.1a ) Y p S (1 ρ )=C α Y α (5.1b ) ìpou S = w 2 w 2 w w [ x π ( x sin wl ejk y + w )] dxdy (5.11) w 2 Diair ntac kat mèlh tic Ex. (5.1a ), (5.1b ) kai lônontac wc proc ρ èqoume, ρ = S 2 k 2 β zk z S 2 k 2 + β z k z (5.12) afoô èqoume antikatast sei ta Y α,y p me tic timèc touc dhlad β z /ωμ kai ωε/k z, antðstoiqa. EpekteÐnontac thn an lush sthn perioq ìpou k x >k, mporoôme na broôme th sqèsh diaspor c, upologðzontac ta shmeða apìklishc tou suntelest an klashc, dhlad brðskontac touc mhdenismoôc tou paronomast tou parap nw kl smatoc. 'Etsi èqoume, S 2 k 2 + β zk z = (5.13) ra k 2 k 2 x k S 2 k = k 2n 2h ( πw (5.14) )2 73

80 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh kai èqoume qrhsimopoi sei ìti kz = k 2 k2 x kai β z = k 2n2 h (π/w)2 ìpou k o kumatikìc arijmìc sto upèrstrwma kai n h o deðkthc di jlashc tou dihlektrikoô pl rwshc twn op n. Parak tw ja parousi soume to di grama diaspor c ìpwc prokôptei apì ta mègista ston q rth tou suntelest an klashc mhdenik c t xhc kai ja to sugkrðnoume me thn kampôlh diaspor c pou prokôptei me to parap nw hmianalutikì el qisto montèlo thc Ex. (5.14). H di taxh pou ja prosmoiwjeð eðnai, ìpwc perigr yame parap nw, di taxh tèleiou met llou me tetragwnikèc opèc kai Ðdia periodikìthta kat x kai y. 'Etsi epilègoume w x = w y =.6L x en prokeimènou na prosomoi soume to peiro b joc epilègoume d =1L x. H prosomoðwsh ja gðnei gia k = (π/L x ) kai k x =...π/l x. Epilègoume N=1, M=, A=1, dhlad 3 ìrouc Floquet kat x, mìno ton jemeli dh ìro Floquet kat y kai apì ènan ìro TE kai TM entìc thc op c. H epilog autoô tou arijmoô ìrwn parèqei kalôterh sôgklish apì to el qisto montèlo kai tautìqrona periorðzei thn apaitoômenh upologistik isqô. To dihlektrikì pl rwshc twn op n èqei dihlekrik stajer n h =3. Prèpei na shmeiwjeð ìti, oi timèc twn L x, L y eðnai ad forec kaj c ìlh h an lush mporeð na gðnei parametrik wc proc autèc. O q rthc tou suntelest an klashc mhdenik c t xhc wc proc ta k kai k x parousi zetai sto sq ma H perioq parat rhshc ja perioristeð dexi apì thn eujeða k x = k pou orðzei th gramm fwtìc tou dihlektrikoô tou uperstr matoc, an to upèrstrwma eðnai o aèrac. ParathroÔme sto q rth, mia seir apì mègista tou suntelest an klashc ta opoða an om dec suneq n kampul n, ìpwc dh èqoume problèyei, prèpei na antistoiqoôn se yeudo SPP rujmoôc. Prokeimènou na belti soume thn eikìna pou èqoume gia touc yeudo SPP rujmoôc, apomon noume ta mègista tou suntelest an klashc kai ta parousi zoume qwrist se nèo di gramma sto sq ma Sto sq ma 5.15 parathroôme ìti sqhmatðzontai di forec kampôlec pou mporoôn na susqetistoôn me diaforetikoôc yeudo SPP rujmoôc. O basikìc rujmìc antistoiqeð sthn pr th kampôlh, aut pou sqhmatðzetai apì tic mple teleðec. H kampôlh aut èqei thn Ðdia akrib c sumperifor me thn antðstoiqh kampôlh tou tupikoô SPP rujmoô pou dhmiourgeðtai sth diepif neia met llou Drude kai dihlektrikoô. Stic qamhlèc suqnìthtec h kampôlh kineðtai pol Ôkont sthn gramm fwtìc tou dihlektrikoô tou uperstr matoc (mplè eujeða sto sq ma 5.15) en kaj c h stajer di doshc aux nei, h suqnìthta proseggðzei mia tim pou stouc tupikoôc SPP onom sthke suqnìthta surface plasmon. Sth sugkekrimènh di taxh h suqnìthta aut isoôtai me π/(1.8l x ), pou sumpðptei me th suqnìthta apokop c tou basikoô TE rujmoô entìc twn op n (k cutoff = π/(n h w x )=π/(1.8l x )). Sto sq ma5.16 ja parousiasteð mìno h kampôlh diaspor c tou basikoô yeudo SPP rujmoô kai h antðstoiqh kampôlh diaspor c pou prokôptei apì to hmianalutikì el qisto montèlo thc Ex. (5.14). Sto sq ma epibebai netai ìti tìso to analutikì montèlo ìso kai h epilog twn megðstwn dðnei thn Ðdia kampôlh diaspor c, epibebai nontac thn orjìthta twn dôo proseggðsewn. Tìso apì to hmianalutikì, ìso kai to arijmhtikì montèlo prosomoi shc, prokôptei ìti kaj c h stajer di doshc tou rujmoô teðnei sto peiro, h suqnìthta proseggðzei thn tim thc suqnìthtac apokop c tou basikoô TE rujmoô entìc twn op n. Parak tw 74

81 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh.6 Χάρτης συντελεστή ανάκλασης μηδενικής τάξης.5 k ( π/ L ) x k( π/ L) x x Sq ma 5.14: Q rthc tou pl touc tou suntelest an klashc mhdenik c t xhc sunart sei twn k kai k x gia di taxh met llou me w x = w y =.6L x d =1L x kai n h =3. To pl toc tou suntelest an klashc èqei logarijmisteð prokeimènou na apeikonðzontai kalôtera ta mègista kaj c se sugkekrimèna shmeða o suntelest c an klashc lamb nei polô meg lec timèc. DiakrÐnontai oi kampôlec twn megðstwn pou antistoiqoôn stouc yeudo SPP rujmoôc..6 Μέγιστα του συντελεστή ανάκλασης και γραμμή φωτός.5.4 kx (π / Lx) k (π/ L x ) Sq ma 5.15: Mègista tou suntelest an klashc ìpwc prokôptoun apì ton q rth pou sqhmatðsthke sto di gramma Sto Ðdio sq ma apotup netai kai h eujeða fwtìc tou uperstr matoc dexi apì thn opoða parousi - zontai oi kampôlec diaspor c ìlwn twn pijan n yeudo SPP rujm n thc diepif neiac. 75

82 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh.7 Καμπύλη Διασποράς.6.5 k (π / Lx ) Αριθμητικός υπολογισμός Γραμμή φωτός υπερστρώματος Θέση μεγίστων συντελεστή ανάκλασης k x(π/ Lx) Sq ma 5.16: KampÔlh diaspor c basikoô yeudo SPP rujmoô, ìpwc prokôptei me efarmog thc Ex. (5.14) kai me thn apomìnwsh twn megðstwn apì ton q rth tou suntelest an klashc. Me kìkkino h gramm fwtìc tou dihlektrikoô, me pr sino h kampôlh diaspor c pou prokôptei apì thn Ex. (5.14) kai ta mple shmeða eðnai ta mègista tou suntelest an - klashc. H sôgklish twn dôo proseggðsewn eðnai p ra polô kal, odhg ntac praktik sthn Ðdia kampôlh diaspor c. ja epiqeir soume, metab llontac ta qarakthristik twn op n (diast seic, ulikì pl rwshc) na epibebai soume ton isqurismì ìti, h asumptwtik proseggizìmenh suqnìthta tou basikoô yeudo SPP rujmoô sth diepif neia tèleiou met llou diamorfwmènou me tetragwnikèc opèc kai dihlektrikoô, isoôtai me th suqnìthta apokop c tou basikoô TE rujmoô twn op n. H pr th perðptwsh eðnai di taxh me w x = w y =.6L x kai n h =1 dhlad Ðdiec diast seic gewmetrðac me thn prohgoômenh perðptwsh, all me kenèc opèc. Se aut thn perðptwsh o kumatikìc arijmìc apokop c tou basikoô TE rujmoô eðnai π/(n h w x )= π/(.6l x ). Sta sq mata 5.17, 5.18 parousi zetaioq rthctou suntelest an klashc mhdenik c t xhc kai h kampôlh pou sqhmatðzoun ta mègista antðstoiqa. ParathroÔme ìti h suqnìthta pou proseggðzetai asumptwtik se aut thn perðptwsh eðnai k sp = k cutoff = π/(n h w x ) dhlad kai p li Ðsh me th suqnìthta apokop c tou basikoô rujmoô entìc twn op n. Sthn deôterh perðptwsh epðlègoume w x = w y =.4L x kai n h =3, dhlad Ðdio ulikì pl rwshc me thn pr th perðptwsh all diaforetik di stash op n. EpÐshc sta sq mata 5.19, 5.2 parousi zetai o q rthc tou suntelest an klashc kai to di gramma diaspor c pou prokôptei apì ta mègista tou suntelest an klashc antðstoiqa. H asumptwtik proseggizìmenh suqnìthta se aut thn perðptwsh eðnai k sp = π/1.2l x = k cutoff. SumperaÐnoume ìti h suqnìthta spoof surface plasmon, kat' analogða thc suqnì- 76

83 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh k( π/ L) x k ( π/ L ) x x Sq ma 5.17: Q rthc suntelest an klashc mhdenik c t xhc sunart sei twn k,k x. Gia th di taxh isqôei w x = w y =.6L x kai n h = k sp=1.666 π/ Lx k (π/ L x) Γραμμή φωτός διηλεκτρικού Μέγιστα συντελεστή ανάκλασης k x (π/ L x ) Sq ma 5.18: KampÔlh diaspor c ìpwc prokôptei apì th jèsh twn megðstwn tou suntelest an klashc. Gia th di taxh isqôei w x = w y =.6L x kai n h =1. 77

84 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh k ( π/ L ) x k x (π / L x ) Sq ma 5.19: Q rthc suntelest an klashc pr thc t xhc sunart sei twn k,k x. Gia th di taxh isqôei w x = w y =.4L x kai n h = k sp =.833 π/ L x.6 k (π/ L x ) Γραμμή φωτός διηλεκτρικού Καμπύλη μεγίστων συντελεστή ανάκλασης k x (π/ L x ) Sq ma 5.2: KampÔlh diaspor c ìpwc prokôptei apì th jèsh twn megðstwn tou suntelest an klashc. Gia th di taxh isqôei w x = w y =.4L x kai n h =3. 78

85 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh thtac surface plasmon twn tupik n SPP rujm n, se èna sôsthma tetragwnik n op n peirou b jouc isoôtai me th suqnìthta apokop c tou basikoô TE rujmoô entìc twn op n. To Ðdio sumpèrasma proèkuye kai apì thn ergasða tou Pendry [17], ìpou akoulhj jhke mða k pwc diaforetik prosèggish apì aut n pou perigr fetai sta pl aðsia thc paroôsac ergasðac. Sthn ergasða tou Pendry apodeðqjhke ìti, èna mètallo, sto opoðo èqoun anoiqjeð tetragwnikèc opèc peirou b jouc, parousi zei isodônamh dihlektrik stajer sth dieôjunsh x kai y, h opoða èqei sumperifor parìmoia me th dihlektrik stajer met llou pou perigr fetai apì to aplì Drude montèlo. Sth nèa aut sqèsh gia thn isodônamh dihlektrik stajer, th jèsh thc suqnìthtac pl smatoc tou met llou èqei p rei, gia to nèo isodônamo ulikì, h suqnìthta apokop c tou basikoô TE rujmoô entìc twn op n. UpenjumÐzoume ìti sthn perðptwsh pragmatikoô met llou, h suqnìthta tou rujmoô gia meg lec timèc thc stajer c di doshc proseggðzei thn tim ω p / 2, ìpou ω p h suqnìthta pl smatoc tou met llou, en sthn perðptwsh tèleiou agwgoô me tetragwnikèc opèc h suqnìthta proseggðzei thn tim ω cutoff, dhlad thn suqnìthta apokop c twn op n Yeudo SPP rujmoð sth dipl diepif neia tèleiou met llou me tetragwnikèc opèc kai dihlektrikoô Sth sunèqeia ja epekteðnoume thn an lush sthn perðptwsh met llou me tetragwnikèc opèc peperasmènou b jouc. H diafor se aut thn perðptwsh eðnai ìti, emfanðzetai plèon h perioq thc met doshc, h opoða kai allhlepidr me thn perioq thc an klashc metab llontac th sumperifor thc di taxhc. Prin proqwr soume me thn prosomoðwsh di taxhc leptoô metallikoô fôllou ja analôsoume sôntoma ti isqôei gia tou pragmatikoôc SPP rujmoôc sth di taxh leptoô metallikoô fôllou me dihlektrikì ekatèrwjen. Sta plaðsia autoô tou kefalaðou, met kai thn an lush pou ègine sqetik me thn exagwg thc sqèshc diaspor c gia thn mon diepif neia met llou dihlektrikoô sto kef laio 2, ja apofôgoume thn pl rh majhmatik an lush gia th eterodi taxh dihlektrikoô met llou dihlektrikoô (IMI Insulator Metal Insulator). O anagn sthc parapèmpetai sto biblðo tou Stefan Maier, Plasmonics: Fundamentals and Applications [22], sto kef laio 2tou opoðou mporeð na brejeð h pl rhc hlektromagnhtik an lush gia touc uposthrizìmenouc rujmoôc apì mia tètoia gewmetrða. Ed apl ja paratejeð to di gramma diaspor c twn uposthrizìmenwn rujm n se mia IMI diepif neia. Sto sq ma5.21 parousi zetai to di gramma diaspor c gia di taxh pragmatikoô met llou, h dihlektrik stajer tou opoðou perigr fetai apì to aplì Drude montèlo qwrðc ap leiec. ProkÔptei ìti up rqoun dôo diaforetik dh rujm n, oi summetrikoð kaioiantisummetrikoð. Parousi zetai to di gramma diaspor c gia dôo diaforetik p qh met llou. 'Opwc parathroôme, oi antisummetrikoô rujmoð èqoun suqnìthta ω + megalôterh apì thn suqnìthta ω sp thc mon c diepif neic met llou dihlektrikoô, en oi summetrikoð rujmoð èqoun suqnìthta ω mikrìterh apì thn suqnìthta ω sp thc antðstoiqhc mon c diepif neiac. Apì to di gramma diaspor c prokôptei ìti kaj c to p qoc tou metallikoô str matoc megal nei oi dôo rujmoð allhlepidroôn el qista metaxô touc kai to di gramma diaspor c proseggðzei to antðstoiqo thc mon c diepif neiac. Sto sq ma5.22 parousi zetai o q rthc tou suntelest an klashc mhdenik c t xhc 79

86 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh Συχνότητα ω [1 15 Hz] αντισυμμετρικοί ρυθμοί ω + συμμετρικοί ρυθμοί ω Σταθερά διάδοσης β [1 7 m -1 ] Sq ma 5.21: Di gramma diaspor c twn summetrik n kai antisummetrik n suzeugmènwn rujm n se str ma met llou pou akoloujeð to montèlo Drude. Parousi zontai ta diagr mmata gia duo diaforetik p qh met llou, me maôrh diakekomènh gramm to p qoc tou met llou eðnai 5nm en gia thn gkrð diakekomènh gramm 1nm. gia di taxh me w x = w y =.6L x kai d =.2L x kai n h =1. ParathroÔme ìti pèra apì ta mègista, emfanizetai kai mia diakrit gramm elaqðstwn tou suntelest an klashc mhdenik c t xhc, pou parousi zei kai aut idiaðtero endiafèron. Gia kal;yterh apeikìnish ja parousi soume sto sq ma 5.23 tic jèseic twn megðstwn kai twn elaqðstwn sunart sei twn k x,k. ParathreÐtai ìti, ta mègista kai ta el qista tou suntelest an klashc sqhmatðzoun dôo xeqwristèc kampôlec diaspor c, pou analogik me thn perðptwsh gewmetrðac IMI pragmatikoô met llou, antistoiqoôn stouc summetrikoôc kai antisummetrikoôc yeudo SPP rujmoôc. H kampôlh twn elaqðstwn proseggðzei th k sp apì p nw ra antistoiqeð stoôc antisummetrikoôc rujmoôc en h kampôlh twn megðstwn apì k tw kai antistoiqeð stouc summetrikoôc rujmoôc. Se epìmenh prosomoðwsh aux noume to p qoc tou met llou apì d =.2L x se d =.7L x. ApeikonÐzoume p li se dôo xeqwrist diagr mmata to q rth tou suntelest an klashc kai tic jèseic twn megðstwn elaqðstwn pou prokôptoun apì autìn, sunart sei twn k x, k. Apì to sq ma 5.24 parathroôme ìti kai se aut thn perðptwsh, sth gewmetrða emfanðzontai tautìqrona el qista kai mègista gia ton suntelest an klashc. Sto sq ma 5.25 apeikonðzoume tic jèseic twn megðstwn kai twn elaqðstwn aut n, ta opoða kai se aut thn perðptwsh sqhmatðzoun tic kampôlec diaspor c twn summetrik n kai antisummetrik n rujm n. 'Opwc parathroôme kajar apì to sq ma 5.25 h apìstash metaxô twn summetrik n kai antisummetrik n rujm n èqei meiwjeð, gegonìc pou epibebai nei ìti kai se aut n thn perðptwsh, oi yeudo SPP rujmoð sumperifèrontai parìmoia me touc antðstoiqouc kanonikoôc SPP rujmoôc pou emfanðzontai sth dipl diepif neia dihlektrikoô met llou dihlektrikoô Sumper smata Sunolik, me b sh tic prosomoi seic pou prohg jhkan prokôptei ìti, tèleiec metallikèc epif neiec, periodik diamorfwmènec me tetragwnikèc opèc mporoôn na upo- 8

87 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh sthrðxoun rujmoôc pou omoi zoun me touc antðstoiqouc thc diepif neiac pragmatikoô met llou dihlektrikoô stic optikèc suqnìthtec. H sumperifor twn yeudo SPP rujm n, ìpwc onom sthkan, eðnai parìmoia me touc tupikoôc SPP rujmoôc tìso sthn perðptwsh thc mon c diepif neiac met llou dihlektrikoô (perðptwsh op n peirou b jouc), ìso kai sthn perðptwsh dipl c diepif neiac, dihlektrikoô met llou dihlektrikoô (opèc peperasmènou b jouc). Sthn perðptwsh thc mon c diepif neiac, h suqnìthta apokop c tou basikoô TE rujmoô entìc twn op n, eðnai to mègejoc pou kajorðzei thn sumperifor thc kampôlhc diaspor c twn yeudo SPP rujm n, kai antistoiqðzetai sth suqnìthta pl smatoc tou montèlou Drude twn tupik n SPP rujm n. AntÐstoiqa, sth dipl diepif neia, emfanðzontai summetrikoð kai antisummetrikoð yeudo SPP rujmoð, me xeqwristèc kampôlec diaspor c, oi opoðec plhsi zoun metaxô touc kaj c aux netai to p qoc tou metallikoô fôllou kai isodônama to p qoc to op n. H sumperifor aut, eðnai parìmoia me th sumperifor twn tupik n SPP rujm n pou uposthrðzontai apì th dipl diepif neia dihlektrikoô pragmatikoô met llou dihlektrikoô. 81

88 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh log( r ) 1.8 k (π/ L ) x k ( π/ L ) x x Sq ma 5.22: Q rthc suntelest an klashc mhdenik c t xhc gia di taxh me w x = w y =.6L x, d =.2L x kai n h =1. EmfanÐzontai tautìqrona suneqeðc kampôlec megðstwn kai elaqðstwn tou suntelest an klashc (kìkkina kai mplè shmeða). UpenjumÐzetai ìti h tim twn megej n L x, L y eðnai adi forh kaj c, ìl ataupìloipa meg jh eðnai ekfrasmèna parametrik wc proc aut..5 k sp μονής διεπιφάνειας.4 k x (π/l x ) Γραμμή φωτός διηλεκτρικού Θέσεις μεγίστων-ελαχίστων του συντελεστή ανάκλασης k x (π/ L x ) Sq ma 5.23: KampÔlh diaspor c ìpwc prokôptei apì th jèsh twn megðstwn kai e- laqðstwn tou suntelest an klashc mhdenik c t xhc apì to di gramma EmfanÐzontai o summetrikìc kai antisummetrikìc rujmìc, ìpwc prokôptoun apì tic jèseic twn megðstwn kai elaqðstwn, antðstoiqa. H sumperifor eðnai parìmoia me ton tupikì SPP rujmì sth dipl diepif neia dihlektrikoô met llou dihlektrikoô (IMI). 82

89 Kef laio 5. Yeudo SPP rujmoð kai exairetik optik met dosh log (r ).5.4 k (π/ L ) x k (π/ L ) x x Sq ma 5.24: Q rthc suntelest an klashc mhdenik c t xhc gia di taxh me w x = w y =.6L x, d =.7L x kai n h =1. EmfanÐzontai tautìqrona suneqeðc kampôlec megðstwn kai elaqðstwn tou suntelest an klashc (kìkkina kai mplè shmeða)..5 k sp μονής διεπιφάνειας.4 k x (π/ L x ) Γραμμή φωτός διηλεκτρικού Θέσεις μεγίστων-ελαχίστων συντελεστή ανάκλασης k x (π/ L x ) Sq ma 5.25: KampÔlh diaspor c ìpwc prokôptei apì th jèsh twn megðstwn kai elaqðstwn tou suntelest an klashc mhdenik c t xhc apì to di - gramma EmfanÐzontai o summetrikìc kai antisummetrikìc rujmìc.upenjumðzetai ìti h tim twn megej n L x, L y eðnai adi forh kaj c, ìla ta upìloipa meg jh eðnai ekfrasmèna parametrik wc proc aut. Shmei netai ìti oi rujmoð èqoun plhsi sei metaxô touc, sumperifor anamenìmenh sômfwna me thn an lush twn kanonik n SPP sth di taxh dihlektrikoô met llou dihlektrikoô. 83