TIME-VARYING PHONONIC CRYSTALS

Transcription

1 IME-VAYIN PHONONI YSALS b D Wan Wgh A hss sub n onfo wh h quns fo h g of Doo of Phosoh h Ewa S. ogs S. Dan of Ea an ou Engnng oaboav Poga wh h Insu fo Boaas an Boa Engnng nvs of oono ogh b D Wan Wgh

2 ABSA -Vang Phonon sas D Wan Wgh PhD Ewa S. ogs S. Dan of Ea an ou Engnng Insu fo Boaas an Boa Engnng nvs of oono oono ON anaa h a obv of hs hss was o gan a unsanng of aous wav oagaon n honon sas aua hos ha nu aas whos os an b va oa n. hs sah was aosh n h was. Fs a D honon sa was sgn a an haaz. Is os os ah hos n hough suaon. h sa onsa ban gas sson an ngav faon. I sv as a ans of uang h aas of honon sa sgn an onsuon an as a hsa vfaon of h o nsng os. N h anssson a ho fo anazng D honon sas was n o nu h ffs of -vang aa aas. h ho was hn us o ov a os-fo souon fo h as of oa -vang aa aas. So ngung sus fo h us of h n ho nu aaa a anssson os an aa afaon. Nw nsghs an b gan fo h govnng quaons an hav h o nf h onons ha a o aa afaon n hs suus. Fna D u sang ho was of o anaz sas wh -vang aa aas. I s shown o b hgh oab wh sng u sang hos. I aows h oa sa f fo a D -vang honon sa o b n. I was shown ha -vang aa aas sgnfan aff h honon sa anssson su an hs was us o swh an nn onohoa wav. Paa afaon an ou un an usans an hs ff was nvsga usng h os-fo souons ov b h nw D ho. - -

3 Absa h o of h n hos gows ogaha as oos na wh sng hos sung n suo ouaona o fo ag nubs of sas. Aso sn boh n hos ov ana souons h a gv fuh nsghs no h faos ha govn h bhavou of -vang honon sas. hs n hos a now b us o sgn an av honon sa ha ou onsa nw o nhan os.

4 AKNOWLEDMENS Fs an foos I wsh o ss os sn gau o Pof. ha obbo a hss suvso. H s asonshng gnous wh hs n guan an fnsh. houghou hs oga h has bo a o o fo boh sona an ofssona an I oo fowa o ou onnu ao. I wou aso o han D. Wan Johnson hss o-suvso. H has ov h vauab sv of ananng a bg-u vw of sah. H has aso gvn an hfu s an suggsons on ovng snaon ss. I hav hoough no ou onvsaons on a va of una os an hs ga sns of huou. Pof. Mohaa Moah an Pof. Aan Nahan hav bn n nough o sv on hss o. h hav oun ov vauab nsgh no nuous ass of sah an h av has ga onbu o h qua of PhD n. Ov h as fou as h bs of Pof. obbo s uasoun gou hav bo fns n aon o oagus. M s o an fo oono wh Mus Muagć an ou nns ng ong an woou sssons ba h hghghs of wow. Af Yu né Wan an Aa annoua h g s no h gou an w ga hna sous. h hav a as sounng boa on ounss oasons an I a v gafu fo ou onnu fnshs an aaa. h a a b Pof. P Buns a Sunnboo has bn gnous wh h qun an s. In aua oss Was an Ahhavan Sushua sn an hous hng o haaz ou honon sa. I an a ga a fo boh of h an of fo h nsv aa n. Ma Wh a fow gaua sun wong wh Pof. Moah was n nough o answ nuous qusons fo ha I su w qu nov fo hs sv. Hs an an h g u o s uh fas han I wou hav bn ab o ahv on own. I a gafu fo hs asssan an guan. D. Howa nsbg who aso o-suvss suns n h uasoun gou has bn a fn an song avoa fo a. H nou o nuosug an aow o s fshan how uasoun vs gh ov a sugon s n n h oang oo. I a aso nb o h fo faang so an nouons o oagus n h f of uasoun sah. - v -

5 Anowgns v I wsh o n hans o un an aun N an Ju Wgh wh who I v wh n own. h ov wh a a o v an a sns of ho an fa ha a vng n wo s ossb an ofoab. I w awas f a ho n 3B. I wou aso o han boh Dav Wgh an hs fané Laua a an boh an ssn-aw Fans an Naa Lo. h nvovn n oono f h baan h ssss of h PhD oga wh fun an aaon an I sha fov b gafu fo h noab s w ha ogh. M ans Wan an Ann Wgh an wf s ans Fans an hs Lo hav awas ha bs nss n n an as suh hav bn suov n ounss was houghou hs n. I wou o anowg h shoash suo I v fo an Onao aua Shoash n Sn an hnoog h ogs Fowsh awa an h onnua suo fo Pof. obbo s Naua Sns an Engnng sah oun of anaa gan. hs suo has bn gnous an s uh aa. Fna an os oan I wsh o a hs hss o wf A. Fv as ago sh suggs ha I gh o w n a PhD oga an ha I shou ons ang. hs was n s of h sona nonvnns wou aus us fo h uaon of h oga. Sh has bn n suov an nouagng an I u wou no hav ahv hs goa no wou f b on n suh a sasfng an fufng on w no fo h ov an suo. Sh s ab an fo f han ou. I wou aso o han ou son Wa fo fng so n n as as I hs hss.

6 ABLE OF ONENS Absa... Anowgns... v Ls of Abbvaons an Sbos... Ls of abs... v Ls of Fgus... v ha Inouon.... Po aas..... Phonon sas Loa sonan Movaon As an so Pvous wo Nonna hoons vang sss ho Av aas....4 Oganzaon of hss... 3 ha Sa Phonon sas n On Dnson Aous wavs n on nson Aous wav anssson an fon n D Po suus Ban gas Dsson ouga ub wavgus Anoaous Do ffs Aaons... 3 ha 3 Sa Phonon sas n wo an h Dnsons Phonon sa ho an ffs Aous wavs n wo an h nsons Ban gas an wavgus Dsson an faon Phonon sa sgn an na sus sa sgn Ena hos sus Sua ha 4 Suaon Mhos h fn-ffn -oan ho Dvaon of a D agoh Inaon v -

7 ab of onns v 4..3 Sab a h anssson a ho Ea auaon Mu sang ho Sang fo on n Sang fo wo ns Sang fo N ns h an wav anson ho Dvaon of h an wav anson ho n 3D Inaon as Oh suaon hos Anab o -vang aa aas ha 5 Dna Phonon sas n On Dnson h -vang anssson a ho anssson hough on vang has vo anssson hough u s onvson ba no a anssson su Souons whn h aa vaaon s o Sng- souon An n- souon h s anssson su os-fo souons A os-fo sson fo on A os-fo souon fo wo an n s Mhooog an vfaon ouaon ho h o of auaon Vfaon sus Paa afaon Sgna swhng Sua an onung susson... 7 ha 6 Dna Phonon sas n wo Dnsons vang u sang ho Sang fo on -vang n Sang fo N oa -vang ns Vfaon On sa n Sva sa ns On oa -vang n Sva oa -vang ns h of funana fqun f Aaon o a -vang honon sa Sua an onung susson... 5 ha 7 Sua an onusons Sua... 5

8 ab of onns v 7.. Pa assuons hss onbuons Sf vons Pa sah onbuons Suggsons fo fuh wo Anaa vons Inaon ovns Ena wo Aaons sah onung as fns... 6

9 LIS OF ABBEVIAIONS AND SYMBOLS Abbvaons D On nsona D wo nsona 3D h nsona AF Av fb oos AO Aouso-o P na ossng un W onnuous wav D D un DF Ds Fou ansfo EAP Eoav o FDD Fn-ffn -oan FF Fas Fou ansfo P ahs ossng un LHS Lf-han s LI Lna -nvaan MF Mao fb oos MIMO Mu-nu u-ouu MS Mu sang ho N Ngav fang PML Pf ah a PWE Pan wav anson P La zona ana HS gh-han s LV oa a vo SAW Sufa aous wav SN Sgna o nos ao MM anssson a ho V-MS -vang u sang ho V-MM -vang anssson a ho - -

10 Ls of Abbvaons an Sbos oan-bas Sbos A A A A A A A A - a a a a a - B B b b b b - b n g φ g n D D Ma of offns n h PWE ho PML offn a Sa anson a Vo of fowa an vs san wav aus n h fqun oan Vo of Fou offns of h nvs of a o -oan funon oss-sona aa of a ub Fowa an vs san wav aus n h fqun oan Vo of fowa an vs san wav aus h h vo of a honon sa a sang Fowa an vs san wav aus PML offn a Funon qu n nng h sang of a -vang n Vo of nna f offns oonas of h n of sa Fowa an vs san wav aus Inna f offn of h h o Mas of Fou offns o of auaon Sffnss offn Eas sffnss nso Vo of sa f offns Vo gou vo Phas vo ou vo agnu Ens of h as sffnss nso Sa f offn of h h o Invs of a o -oan funon h h Fou offn of h nvs of a o -oan funon Vo of nn f offns Da

11 Ls of Abbvaons an Sbos E E F F n f f f f H I I n I J Dsan K L M M M M PML ang funons Inn f offn of h h o PML offn as Vo of Fou offns of a o -oan funon Fo a oson n Fqun Pung fqun v fqun ans fqun Ma qu o fn h Fou offns of h nvs of a o -oan funon oa a vo na Han funon of h fs n an h o In a Inn f nns Pu f nns An ng na Bss funon of h fs n an h o An ng Bu ouus Wavvo An ng Sng onsan Wav nub An ng Lngh hough a honon sa A on n h fs Boun zon Invs of a o -oan funon Mau FDD n n h -on Mau o of h na wavfunons ons M M M Vaabs n h sab anass of an FDD suaon M h h Fou offn of h nvs of a o -oan funon

12 Ls of Abbvaons an Sbos N N An ng Mass Mau FDD n n h -on Nub of ns n a u sang onfguaon N N N Vaabs n h sab anass of an FDD suaon N N N N z n P P Nub of aa ons o wavfo Nub of ons n h o z nsons An ng Invs of h -vang ns Monu P P P Vaab n h sab anass of an FDD suaon q An ng An ng aus fon offn. Poson vos S An ng Sang a S Ŝ Saaon as S S S oss-sona aa h h n of a sang a h h n of a saaon a S n h n h n of a -vang saaon a S n S n s P n Sss n h h nson u o san n h n h nson h n- anssson offn An ng anson a anssson a anssson a Poagaon anssson a anssson offn Wav ans hough a aa wh a -vang has vo h h n of a anssson o a h n h n of a anson a

13 Ls of Abbvaons an Sbos n h n h n of a -vang anson a u u u Ma of gnvos n h PWE ho Aous a vo san f Longuna aous a vo san f ansvs aous a vo san f u Aous a san u Saa vo ona fs u n V V v v v W w w X X n Y Aous a san n h h nson Aous a san a oson n Ma qu n nng h sang of a -vang n Vou PML sa vo Vo ona ns a nab na sa Vo of a fo Ma o vo qu n nng h sang of a -vang n PML sa vo A on n h fs Boun zon oa FDD suaon san n h -on Poson vo -Bas Sbos α α α β β Γ z Poson n h n h nson oa FDD suaon san n h -on haas aous an h s of a ngs Da a of h Fou offns of h nvs of a o oan funon PML ang offns Vaab n h sab anass of an FDD suaon offn n h -a A on n h fs Boun zon souon n h -oan souon n h o z nsons

14 Ls of Abbvaons an Sbos v δ( ) Da a funon δ n η θ θ n θ ou κ Λ λ λ λ Kon a Fng faon An ang Phas nng an ng a gon Aaba ossb Ma of gnvaus Wavngh Egnvau Fs Laé offn µ Son Laé offn µ Lna ass ns ρ σ φ φ Dns Sss nso Saa vo ona Phas ang ψ ψˆ Ougong an gua na wavfunons of o ψ ψˆ Vos of ougong an gua na wavfunons An angua fqun Pung angua fqun Msanous Sbos Funons an Abbvaons Hang san Foo oaon I { } Fou ansfo I { } Invs Fou ansfo ~ Ovso nos vaabs a o nua sson ^ Ovso a nos a un vo * onvouon o o onugaon Ba ovso nos o onugaon Sus nos h a ansfo

15 LIS OF ABLES ab 3.. ansu a aas ab 3.. Hos an sa os ab 3.3. sa a aas ab 3.4. o aous f aas... 5 ab 4.. auab suaon gons an o bounas... 6 ab 4.. onnuous PML absobng bouna quaons ab 4.3. Ds PML absobng bouna quaons v -

16 LIS OF FIES Fgu.. h oang g of o aas... Fgu.. On wo an h nsona honon sas... 3 Fgu.3. A wo nsona honon sa... 4 Fgu.4. h sa f of wo aousa-ha ns... 4 Fgu.5. An usaon of h o of sonan n aaas... 5 Fgu.6. wo a onfguaons of a nonna hoon sa... Fgu.. a) A fu-f g ub an b) s aoaon as a na han of asss an sngs... 5 Fgu.. Inn ans an f wavs a an nfa... 7 Fgu.3. h nonau fo a wo-oonn on nsona honon sa... 8 Fgu.4. onsuv an suv nfn of ans an f wavs... 9 Fgu.5. Sua ban gas a of a D honon sa... 9 Fgu.6. Ban ga haass fo vaous an aos... Fgu.7. h anssson sa fo vaous fng faons (η)... Fgu.8. h anssson sa fo vaous has vo aos... Fgu.9. h anssson su fo onguna wavs n an aunu-n honon sa... Fgu.. Iusaon of unnng hough a sng ba... 3 Fgu.. Iusaon of os aow n an nfn oub ba... 4 Fgu.. h sson aon a) an gou vo b) fo a s o ss... 6 Fgu.3. h sson uvs fo an aunu-n honon sa... 7 Fgu.4. A ouga ub wavgu... 8 Fgu.5. A sgn of a D o aous wavgu onanng wo sons... 9 Fgu.6. gua a) an anoaous b) Do ffs v -

17 Ls of Fgus v Fgu 3.. Eas wav oagaon n wo nsons Fgu 3.. Aous wav sang off an nfn ns n n wo nsons Fgu 3.3. h wavvos a h nfa bwn wo soo a Fgu 3.4. h qufqun uvs us o aua faon a an nfa Fgu 3.5. wo ss of qufqun an osonng sson uvs Fgu 3.6. h sson uvs fo a D honon sa... 4 Fgu 3.7. Wav faon un noa an ngav fang onons... 4 Fgu 3.8. An usaon of fousng n an N sab... 4 Fgu 3.9. h fqun sons fo h ss of ansu as Fgu 3.. o vw of h honon sa sgn Fgu 3.. o a) an s b) vw of h faba honon sa Fgu 3.. Ena onfguaon Fgu 3.3. Masu an sua anssson su of h honon sa Fgu 3.4. Masu an sua sson uvs of h honon sa... 5 Fgu 3.5. h oa an ga-on sgnas... 5 Fgu 3.6. A aga of h na onfguaon a) an h asu aous f.. 53 Fgu 4.. h offs g naon of h FDD sa vaabs... 6 Fgu 4.. FDD agoh fow fo o bounas... 6 Fgu 4.3. Nonau us o sb san wavs a h bounas of a D sa Fgu 4.4. aua anssson su fo a D ub wavgu Fgu 4.5. h onsuon of h fs ub Boun zon fo a angua a Fgu 5.. A onoon s sa an oua a v -vang nfa... Fgu 5.. A onoon s oua as oagas hough a -vang sab of aa Fgu 5.3. h -vang anssson a... 6 Fgu 5.4. A fowha usang h ss an quaons nssa o sua wav oagaon hough a -vang ub wavgu... 7

18 Ls of Fgus v Fgu 5.5. A o of h nub of ngaons qu fo n -vang s... 9 Fgu 5.6. oason of anssson sa usng h nw ho an a FDD ho... Fgu 5.7. Funana fqun anssson offns fo n sa an -vang s... Fgu 5.8. Anoh funana fqun anssson su fo n sa an -vang s... Fgu 5.9. anssson sa hough n -vang D honon sa s... Fgu 5.. A sufa oos of h funana fqun anssson sa fo vaous ung fquns... 3 Fgu 5.. h anssson sa fo wo ffn ung fquns... 6 Fgu 5.. h noaz sho- Fou ansfo of a Hz ans sgna... 7 Fgu 6.. h -vang sa f offn vo Fgu 6.. A s -vang -a Fgu 6.3. h o FDD sou au on h au an... 4 Fgu 6.4. h sua san fs fo a sng sa n... 4 Fgu 6.5. h sua san fs fo h sa ns... 4 Fgu 6.6. h noaz san sa of u z fo a sng n Fgu 6.7. h sua funana fqun san fs fo on -vang n Fgu 6.8. h sua san f haons fo on -vang n Fgu 6.9. h noaz san sa of u z fo h ns Fgu 6.. h sua funana fqun san fs fo h -vang ns Fgu 6.. h sua san f haons fo h -vang ns Fgu 6.. h hang n sa san f agnu Fgu 6.3. A a) sa an b) na honon sa Fgu 6.4. h san f haons sn n h sa f of h na honon sa... 49

19 ha INODION Po aous aas a honon sas a of ns baus aous wavs bhav ffn n h han n hoognous aas. hs ngn aas hav bn su fo ov a nu an h bo of ho ha sbs h s nsv. h ofn aab os hs sas hb a s b h ho of onsun aas an h sz an sang of hos aas ov a h sgn aas of h sa. n now h os of honon sas w f an unhangab baus h sgn aas w f an unhangab. Howv h an s of av aas ha ou b us n a of assv aas n honon sas. Av aas hav ansuv os an h aa aas an b hang b ang an na suus. Suh av aas n aon o s hana ans hav bn us n honon sas o hang h sgn aas fo on sa sa o anoh hus nabng av sas o hav o han on o of oaon. hs hss has an h on fuh an ss o an h suaon whn h sgn aas of a honon sa a onnuous hangng n. h ho ha sbs sa honon sas s n o han -vang aa aas an ons bas on h n hos a sn. hs ha nous o aas n gna an sns so of h aabs. Sa anon s a o honon sas as h a a h ha of hs hss. h ossb of nooang oa sonan sas n honon sas s aso suss baus hs as o h su of aaas. h ovaon fo usng av aas n honon sas s hn an nung h as an so of h hss. A boa vw of h vous wo aosh n hs f s off fousng on h wo n nonna hoons -vang sss ho an av aas. Fna an ovvw of h an of h hss s gvn. hs ha onans aa ubsh n h 9 a b DW Wgh an S obbo aang n asoun 7() n h haass an aaons of aaas. - -

20 ha. Inouon. Po aas Po aas a ngn ooss ha hb unqu os no usua foun n nau. Ha Po- nvsb oang [] bnng wavs n h wong on [] an subwavngh agng [3] a us a fw of h o susng aabs. Po aas an b sgn o anua aous [4] oagn [5] ha [6] ason [7] o us abou an oh n of bu o sufa wavs of ns n nw an ona oan was. h ossss suh unqu abs baus h hav haas fau szs ha an b uh ss han o on h sa o as h wavnghs of ns. No on os h ho of onsun aas aff h aoso os of a o aa bu h sa-sa sz an suu aso a a sgnfan o whh sngushs h fo gua aas. Po aas onssng of a hos f wh sas a a hoon [8] o honon [9] sas nng on whh h sa oagn o aous wavs sv. In hs sas h fau szs a sa o h wavngh whh as o song nfn ffs fo sang off of h afa sa a. Effv aa aas a no w fn n hs suus. Insa on onss h aonsh bwn saa an oa fquns a h sson aon. h fous of hs hss s on honon sas. Fgu.. h oang g of o aas. Phonon sas hav fau szs a o sa han h wavnghs of ns. Whn h wavngh s ag oa o h fau sz of h honon sa wav oagaon an b haaz usng ffv aa aas an h aa aas o b hoognous (a a aaa). As h wavngh aoahs h fau sz sang ffs bgn o ona an h sson os of h sa o no a.

21 ha. Inouon 3 Po aas whos fau szs a uh sa han h wavnghs of ns onss of a hos f wh s sonaos an a ofn a aaas. h wavnghs of ns a uh ag han h nvua sonaos so h wav os no s h. ah bhavs as f avng hough a hoognous u wh a aua s of ffv aa os as shown n Fgu. an hos ffv os an b aus o vaus s b unng h sonaos. hs ffv aa os an b v ag v sa o vn ngav whh an a o so nsng os an aaons. h snon bwn sonan an sang o aas s no abu an qu ofn suus a fa no boh agos []. hs suus a a oa sonan honon o hoon sas. In hs suus ah sa s aso a sonao un fo an fquns. A sho wavnghs h sson aon a sb ha of a u sang honon o hoon sa bu a ong wavnghs a hav aona bans sn u o h oa sonans of h sas... Phonon sas Phonon sas a aas oos of gua sa aous sas. h an b on wo o h nsona as usa n Fgu.. A wo nsona honon sa a b ou gou s shown n Fgu.3. h ob of wav oagaon hough suh a aa has bn ons sn h -nnnh nu [] an sn hn h ogss n onsng v o oa sang suus has bn onnuous. An ona a an a aoun of h vaous aoahs o nng wav oagaon hough ffn sang suus has bn sb n a n boo b Man []. Fgu.. On wo an h nsona honon sas. In on nson (D) h sa onsss of nfn ans of anang aas sa ogh as n a Bagg fo. In wo nsons (D) h sa s a u of nfn os gua sa an b n a hos u. In h nsons (3D) h sa s oos of s sas susn n a sa foaon whn a hos u.

22 ha. Inouon 4 Fgu.3. A wo nsona honon sa onssng of vn ows of /8 sanss s ns a n a squa aa an susn n a wa hos. h fng faon s.58 anng ha 58% of h vou whn h sa s sanss s. hs sa hbs ngav faon aoun 4 Hz baus of h afu sgn sha of h sson uvs. h s an o nau of honon sas bos nasng oan as h wavngh aoahs h sa sz sa as h onsun sas. Effv aa aas a no ong as fn an nfn ffs o no a. h sz an sang of h sas bos nasng van. As an a ons h sa aous f aus b h wo soun-ha ns shown n Fgu.4a) fo an nn wavngh na o h n a. Song nfn ffs an b sn as as an nus of ng aang awa fo h ns. In Fgu.4b) h sang bwn h ns has nas b us % an now h sa f s aos o ffn onsang how oan h fau sz an sang whn onsng sang ffs n honon sas. Fgu.4. h sa f of wo aousa-ha ns a b an nn an wav. h nn f (no shown) has a wavngh na o h n a n boh ags. In a) nfn ffs an b sn as as an nus n h sa f. In b) h ns hav bn saa b an aona % an h sa f s aaa ffn nang h oan of sa sz an sang.

23 ha. Inouon 5 So of h nsng ffs no noa foun n nau ha a onsa b honon sas nu anssson ban gas [3] ngav wav faon [4] uasoun fousng n wo [9] an h [5] nsons unnng [6] anoaous Do ffs [7] an subwavngh agng aabs [8]... Loa sonan Phonon sas wh hoognous sas a o no hb sonan ffs a ong wavnghs an hus h os as usv fo nfn an sson ffs. Howv f h sas onss of u aas sgn o sona a aua wavnghs hn h honon sa a bhav as a sonan aaa a ong wavnghs an as a sang sa a sho wavnghs ([] [9] [3]). hus hs oa sonan honon o hoon sas an b vw as hoognous sonan suus fo ong wavnghs an as s sang suus fo sho wavnghs. An ffv u ho s us o sb h haass of h aaa n s of ffv aa aas whh s no ossb fo h sho wavnghs ha aus sang ffs n honon sas. Fgu.5. An usaon of h o of sonan n aaas. sonan an b us o ahv ngav ffv aa aas as shown abov fo a nuu. A ow fquns h ass ovs ogh wh h a fo. Bu as h fqun nass h ass vnua ss ou of has an ovs n ooson o h a fo. hus h nuu hbs a ngav ffv ass a ha aua fqun. In h hoognous whn h wavngh s uh ong han h sonao sz ffv aa aas a w-fn an n no on on h onsun aas bu aso on h sonao os. h ffv aas an b aus o vaus b unng h subwavngh sonaos. Po un hs sonans an a o ngav ffv os. A s wa o unsan hs on s o nvson a nuu

24 ha. Inouon 6 onssng of a ass an a sng an bng ush an u abou hafwa u as shown n Fgu.5. A ow fquns ng s a o h nuu an ovs n h on of a fo. Howv as s ush an u fas an fas h nuu bgns o swng ou of has wh h a fo. Evnua h a fo s ushng fowa wh h nuu s ovng bawas. In ssn h nuu has a ngav ffv ass a ha fqun sn s ovng n ooson o h on of fo. Aous aaas hav bn osua an onsa bas on hs ns. Aousa a 3D sonan aaa an b a b bng onn shs of ffng aas no a hos u. Fo a ubb-oa sanss s shs an b un so ha a a aua fqun h nn sh s ovng n ooson o h ou sh ang o ngav ffv vaus of ns an ossb. Mo a nwo of asss an sngs ou b a o ahv h sa ff. Aous quvans of ngav faon [4] oang [3] an subwavngh agng [33] hav bn osua an n so ass onsa na.. Movaon Phonon sas a noa sa n h sns ha h os a f n avan b h sgn aas whh an h funona o v sf aangns. So fa usfu hnona n honon sas a b h fa ha h ffs a on aab ov ah naow fqun angs. Fo a h honon sa sb n [5] was sgn o fous uasoun a.57 MHz bu whn h fqun was hang o.6 MHz (a.9% hang) h fousng ffs w no ong aan. As anoh a h ban ga wh was shown n [3] o b hgh nn on h sa fng faon. hs onsans aa o h usfunss an vsa of honon sas o sgn naowban oang onons. W hav oos ha a b ossb o naa a h bhavou of honon sas b hangng h aa aas as a funon of [34]. Vang h aa aas of h sas w of h oagaon of aous wavs hough a honon sa. hs a nab -vaan honon sas o hav o obus o u oang gs as oa o h sa ounas. On aoah o nng h ffs of -vang aa aas s hough h us of h fn-ffn -oan (FDD) ho whh s a oon an vsa aoah o aousa suaon [35] an s suss a ngh n ha 4. Whn a suaon gon h sa oans a sz an h aa os whn hs gon a sn on hs s g. h onnuous aous wav quaons a aso

25 ha. Inouon 7 sz an us o aua h aousa fs ov nna ss. h sz quaons a anab o -vang aa aas wh o no ofaons whh as h FDD ho an aav sang on fo h anass of -vang honon sas [36]. Howv on ao awba of h FDD ho s ha gvs no anaa nsgh no wh a aua su ous on wha ous. Moov baus h ho s nhn an aoaon of h govnng quaons h aua ns on h souon n boh h sa an oans. hus b sus a a uh ong o gna han wh an a ho. o ass hs shoongs w hav us h on nsona anssson a ho (MM) as an anav sang on sn gvs a os-fo souon o wav oagaon n o a [37]. In wo nsons w hav us u sang ho (MS) sn s anaogous o h MM ho []. W n hs hos o han vang aa aas whh souons o aous wav oagaon n vang honon sas ha a a n n... As an so As h ua goa of hs hss s o sb wav oagaon n -vang honon sas. h aa ouos of hs sah ou b h von of nw agng hos wh ov souon aas whos ansv an fv an b aus n a- an aas wh av ang os. As of hs suus hav no bn a; howv hs hss as h gounwo fo fuh oaons no hs o. o unsan -vang honon sas w hav n h sng sa honon sa ho so as o han -vang aa aas an aso o ov nsgh no h faos ha govn h bhavou. W hn anaz h sung govnng quaons o gan nsgh no how o ono hs aas. h os-fo ssons a qu h an gv nsgh no uosfu ono of h wav anssson os of -vang honon sas. So hs hss fouas h bas ho nssa o unsan h os of -vang honon sas. A sa honon sa was a o vf ha ou sa hos ah In hs hss w a os-fo o an an quaon onssng of a una s of s. h a souons onan nfn an s an a hfo no os-fo.

26 ha. Inouon 8 h aqu na aa. hn on ou na hos w a h w anaz o nsu ha h oas o h sa ounas un sa onons. In hs wa a na h was a uon ou nw hos o a su ha h a va n h sa as. h onsuon an sng of a -vang honon sa s no whn h so of hs hss sn was ossb o ngb sgn suh a sa whou a ho of how o o so. ha ho s h onbuon of hs hss an s f as a fuu o o onsu an haaz a na sa usng h nw ho. h s fuh susson gang how a a-wo -vang honon sa gh b onsu an oa. Possb ffns bwn h ons of h ho n hs hss an fuu na sus s aso suss..3 Pvous wo Sus n h 96s o 97s hav an -vang aas n oagn sss [38] [46] bu non gv aqua nsgh no h s of sss wh whh w a onn. Suh ffos hav nu hoognous -vang a [38][39] -vang ough sufas [4] nfn sa wavvo aoahs [4] an os noab wo on oagang -vaan aas [4][43]. -vang sang was ons fo ovng sas n [44] [46] bu no nu h ffs of -vang aa aas. A aua nsng anass of a -vang Bagg fo has bn sn b Wu a [47] howv s vaon os no o aoun fo h au ouaon of h -vang an sonnus. In hoons Wnn a [48] show ha a -vang v an oua ng fo on hoon ng ban o anoh n an nban anson oss. h sson os of a sa hoon sa w n an hn h ff of ubng hs ss b vang h onsan was an o v h unng ahas whh was hn vf wh a nua suaon. In aouss h hav bn so n sus n onfguab sss whh a quassa n h sns ha a aua aa s hang an h os of h honon sa a -asu [49] [59]. In [49] an [5] h sas a squa os ha a hsa oa o aus h honon sa os. A aso s us as an av aa n a D honon sa n [5] an a D honon sa n [5]. In [53] [59] h ho an so na aa fo usng vaous av aas as ausab sas n D an D honon sas s sn.

27 ha. Inouon 9 hs sus a oan baus h haas h nvua sa oang ons of a vaab aa ss bu h o no au h ffs of vang hos aas n. h aso onsa h vau of usng na aa aas n honon sas b usang suh ffs as nas anssson o anuaon an a ban gas an sson. hs hss ns hs sus o ass wh h aa aas a hangng as a funon of. I w b sb how -vang aa aas sgnfan aff h honon sa anssson su bon wha s ossb wh s ausab sas an how hs was us o swh an nn onohoa wav. Fuho h os-fo souons va h oun of an onons nssa fo aa afaon..3. Nonna hoons So aas hav a nonna sons o h nn f a hgh f nnss. a h a wo onfguaons us n nonna oa sss as usa n Fgu.6 fo a nonna hoon sa. h fs s on h nns of h nn gh o oua h aa os of h nonna shown n a). h son s on a saa ba whos so uos s o oua h aa os of h nonna ofn a h u as n b). gass of h onfguaon f h aa s aso o hn bhavs as a nonna hoon sa. [6][6] hs son onfguaon bas h oss sban o h -vang honon sas howv h a so noab ffns. In h -vang honon sas sb n hs hss h ung hans s ono b a f ha os no na wh h aousa f. Fo a an oagn f gh b onong h aa aas whh wou no na wh h aousa f. hus h nns of h aousa f s van whn asonab s. In a u nonna hoon sa h nns of h nn wav aso ouas h aa aas an a o no a unss s agnu s ow nough o gno s onbuon. If h nns of h aousa f s hgh nough o a nonna sons n h aa h ff s funana ffn han h ouaon aus b ung. Suh ffs a no ons n hs hss.

28 ha. Inouon a) Inn (I n ) (I n ) (I n ) (I n ) (I n ) b) Inn (I n ) (I ) (I ) (I ) (I ) Pu (I ) I >> I n Fgu.6. wo a onfguaons of a nonna hoon sa. In a) h nns of h nn f (I n ) suas h nonna sons of aa. In b) a saa u f suas h nonna ff. h u f s uh o nns (I ) han h nn f so ha h nns of h nn f os no sgnfan aff h nonna aa aas. h oh ao ffn s ha h fquns of ns n h -vang honon sas a os of agnu ow han n a nonna hoon sa. In ou aousa sss h aa aas a b vang a o vn abov h nn fquns. h na aas an a o h ung f on a sa uh sho han on o of h nn wav. In an oa nonna hoon sa h aas son o h nvo of h ung sgna as oos o s nsananous vau an so h hang n aa aas s ons sow vang as oa o h nn fquns. hs assuon s no a n ou subsqun anass..3. -vang sss ho h bhavou of na sss wh -vang aas has bn vous o [6]. h H quaon asss son-o sss wh oa vang aas an s assa sa as ( a qψ ( ) ) & (.)

29 ha. Inouon wh s a sa vaab suh as h aous a san o h f a s h D offs of h -vang aa q s h agnu of h vaaon an ψ ( ) s a o funon wh un au. h os w us fo of hs quaon s whn ( ) os ψ whh as o h Mahu quaon: ( a q os ) & &. (.) hs quaons hav bn su nsv an hav su n nuous aaons suh as aa afs [63] an aaons [64] anqua ga soag vs ha hav n foun a nw nh [65]. hs quaons hav aso bn a o sa hoon sas whos aas a o n sa nsa of. In hs as ( ) sgn( sn ) ψ whh s a squa wav wh sns sa nsa of. hs subsuon no h H quaon s a h Mssn quaon: ( a qsgn( sn ) ) & &. (.3) Anass of hs saa o ss vas h fqun ass an so bans an sson uvs ha on wou fo a hoon sa [6]. As aav as hs quaons an h aoanng bo of ho a h on a wh o n on nson h sa o bu no n boh suanous. hus h ho of oa -vang sss n s un fo s unsuab fo anazng vang honon sas..3.3 Av aas a n hs susson has bn ha h a av aas aab of bng us o onsu -vang honon sas. In fa h a nuous aas usng a va of ansuon hanss ha ou b us as -vang sas. H w sn a fw of h o noab anas. h subsqun nfoaon s no nn as an n-h o hausv snaon of av aas bu ah s nn o show ha a a of av aas ss fo sgnng a -vang honon sas. h foowng hnoogs a so of h nws an os osng av aas ha ou b us n -vang honon sas Mao an av fb ooss Pzo aas n a san n h sn of na a f (h nvs zo ff). hs ff an s ua h zo ff s wh zo aas a us n uasoun ansus [66]. Howv n s of h ab o fo n an f h us n av aas s ha b h bnss of

30 ha. Inouon zoa aas oo onfoab (aua whn a o uvna sufas) soo san auaon an ova ow san ng ns [67]. o ovo hs unsab haass oos aas hav bn a onssng of sanwhs of zoa fbs b n o an ov wh nga os on h sufa. If h fbs a squa n oss-son hn h aa s a a ao fb oos (MF) [67]. If h fbs a ua hn h aa s a an av fb oos (AF) [68]. hs aas hb nas auaon san ov onoh ansu zo aas suh as P an an onfo o vaous uv sufas [69]. Eoav os h a an os a oav os (EAPs) whos sha s a un h nfun of an f. Of h nuous vas of EAPs uh of h fous has bn on asos [7]. h a nuous sf s of asos an an hav sab quas suh as bng ghwgh havng ag auaon san an havng h aab of gnang a usfu aoun of fo. hs aas hav vn bn a o a wo nsona honon sa as was vous non [5] an hus a a goo ana fo a na sang aa. Magnosv ooss h agnosv ff ous n an aas whn an na agn f auss h agn oans o hang h sha. Baus of hs ff h aa an onga n h sn of an a agn f [7]. n so of h obs assoa wh aona agnosv aas suh as bnss an osss u o uns hav bn ovo b ang a agnosv-o oos [7]. hs ooss aa o b uh o uab hav b sons an o no saf h san foan of u agnosv oouns. Fo sha o aos Sha o aos an b fo an hn s o h ogna sha b hang h whh auss h o ungo a aa has anson ba o h ogna sa. h hav a ag auaon san bu qu o b ha an oo an hus a sow. On h oh han agnosv aas an a qu bu o no show na as uh auaon san. A nw of aa a a foagn sha o ao s a sha o ao whos has anson s aus b an a agn f nsa of b ha

31 ha. Inouon 3 [73]. hus hs aas show h ag auaon san of a sha o ao an h fas aon of a agnosv aa. abon nanoubs An aog a of uwa abon nanoubs has n bn vo ha s gh has v ag auaon san an oa a v hgh fquns has a v ag oaona au ang an a has v hgh wo aa [74]. hs nw aa ss o obn a h os sab quas of an av aa an a bo v vasv n sss qung av aas..4 Oganzaon of hss hs hss s oganz as foows. ha nous sa honon sas an h os n on nson. ha 3 ns hs ons o sa honon sas n wo an h nsons an os o o ssus suh as wav sson an faon. h na sus fo a D honon sa ha w anufau a nu n hs ha. A va of hos fo anazng honon sas an a susson of h snghs an wansss s sn n ha 4. h sa ho s n o han -vang aa aas n on nson n ha 5 an s n o wo nsons n ha 6. h bu of nw sah onbuons n hs wo has. A sua an so onusons of hs hss aong wh suggsons fo fuu sah a off n ha 7.

32 ha SAI PHONONI YSALS IN ONE DIMENSION hs ha os aous wav oagaon hough honon sas n on nson. Fs h D aous saa wav quaon s v. fon an anssson hough h nfas bwn ffng aous aas a hn suss. N o aous suus a nou aong wh so of h a anssson an sson os. A ouga ub wavgu s sn as on of honon sa an so of s usfu os a o. Fna so aaons of D honon sas a sb.. Aous wavs n on nson In hs son w v h D aous saa wav quaon usng an aoa o of a fu-f g ub. ons a g ub of oss-sona aa A f wh a ossss hoognous fu u. h fu has a ns ρ an a onguna ouus. W ons h as wh h oon s onsan o on nson aong h n as of h ub. hus w an fn h na ass ns as µ ρa. ngh o aoa h oon of h fu n h ub w v h ub no sa sons of. As a fs-o aoaon w sn ah of hs sa fu sons as a ass wh a vau of µ an a sng wh a vau of A. Fgu.a) usas h fu-f g ub an Fgu.b) shows h osonng ass-sng aoaon. h a s n ha hs s a u afa onsu ha s nou fo onua uoss aon. wo onons us b gang h sz of h fu sgn: h fs s us b uh ag han nvua ous an aos so ha h unng s nau of h aa an b nasua n ffv aa aas suh as ns an onguna ouus. h son s ha us b uh sa han h aous wavnghs un onsaon o avo h ffs of sson as w b suss a. hus h aa an b vw as a onnuu of fu sgns wh unfo ffv aa aas whh w hav aoa as asss an sngs. Mhos fo anazng wav oagaon n honon sas a f o ha 4 u o h so of hs o

33 ha. Sa Phonon sas n On Dnson 5 Fgu.. a) A fu-f g ub an b) s aoaon as a na han of asss an sngs. In h aoaon h asss a onfn o onguna oon on. In hs onfguaon w ao a Lagangan fn fa so ha ( ) sbs h san fo qubu of a ass a oaon an a as oos o s absou oson n sa whh s an Euan fn fa. ha s ( ) ( ) an Lagangan Euan a qubu. h asss an b fn usng an n n wh n s an ng an osons o h ass oa a oson h san ( n ) h n h fu son. n. Aong n ( ) fs o whh s h san fo qubu of h n of ass of h asss n fos u o h san an ha of h nghbous. Aong o Hoo s aw h fo a a oson n F n F n s ( n n ) ( n n ) ( ) n n n (.) wh h nn of an F hav bn ov fo a. Nwon s son aw sas ha F n n. (.) If h ns an sffnss a assu o b oa onsan h wo quaons an b ou as whh an b ss o a as A n ρ A ( ) n n n (.3)

34 ha. Sa Phonon sas n On Dnson 6 n ( n n n ) ( ) (.4) wh ρ s h onguna has vo. B ang h fao on h gh han s of (.4) aoahs ( ) whh an b sn b nong ha h gh han s fao s h na-ffn aoaon o h son vav. hs s a va aoaon ov ha s uh sa han h wavnghs of ns an uh ag han h nvua onsun aos an ous of h fu. In hs aoaon h quaon bos h on nsona saa wav quaon n a hoognous fu u: ( ) ( ) Fo -haon oon of a aua angua fqun saa an oa oonns as ( ) u( ) { }. (.5) ( ) an b saa no (.6) an hus a ow-as u nas h saa vang oonn of a onohoa wav of angua fqun. on nsng (.6) no (.5) h wav quaon us o h on nsona Hhoz quaon u ( ) (.7) wh s h wavnub. Souons an b wn as an wavs of h fo { } ± ( ) ( ). (.8) h ho of sgn n (.8) s aba bu h onvnon hosn w b sa wh aoa. In vng hs s souon o h wav quaon w hav a a nub of assuons na ha h sa-sgna aoaons a va [66] an ha h u s fu hoognous soo an ossss. h ssu of h va of hs assuons w b suss n on of h obs an souons sn n subsqun has. h D saa wav quaon a aso b v bgnnng fo a onnuu son of a hoognous u an an b foun n an s suh as [75].

35 ha. Sa Phonon sas n On Dnson 7. Aous wav anssson an fon n D ons aa o b hoognous wh ns an bu ouus ρ an K sv. a ha K κ wh s h soo as sffnss offn (Vog noaon [75]) an κ s h aaba ossb [66] a of aa. Fuh s ossss an ssonss has a has vo of an a haas aous an of ρ (.9) K ρ K (.) whh fo bv w subsqun b f o as h aous an. Aous wavs sa whn h ng uon a bouna bwn ffng aas an as usa n Fgu.. wo onons us b a h bouna: h ssus on h s us b na an h noa oonn of h vo us b qua. If h of hs onons s no hn h w b h a ssu o vo sou a h bouna. In on nson hs wo onons an b an a au b anssson an fon offns whh fo a onohoa wav an b ss n s of h haas ans of h wo a. Fgu.. Inn ans an f wavs a an nfa. onsng san wavs h anssson offn fo aa no aa s an h fon offn s Fo h san wavs un onsaon ( ) (.) ( ) ( ). (.). (.3)

36 ha. Sa Phonon sas n On Dnson 8.3 Po suus As sb n Fgu. a honon sa n on nson onsss of nfn ans of anang aas sa ogh. h anson bwn aas n no b abu an h sa a onss of an ffn aas ov ha h suu s o. Fgu.3. h nonau fo a wo-oonn on nsona honon sa onssng of a sa of nfn ans of anang aas. h nonau us fo a wo-oonn honon sa onssng of a sa of anang ans s ov n Fgu.3. h nu ngh a whh h suu s o s a h sz a. Fo a wo-oonn ss a aa a h fng faon η fns how uh of hos aa s a wh an nuson. In h fgu aa s h hos an aa s h nuson. hs s aangn of anang aas as o so noab anssson ffs suh as ban gas unnng an sson a of whh a bf sb n h n wo subsons. sah on honon sas has bn a n D an 3D suus howv h has bn so anass of u D sss [76] [88]. h anssson a ho was sb n [76] an subsqun ns onsa ban gas n D honon sas [77]. A u hoa anass of o on nsona sss s gvn n [78]. A han of asss was o na n [79] an onsa h hoa ban gas. Wavgus suus ha fo wavs o oaga n aua ons w anaz n D n [8]. Bagg fos onssng of anang ans of aa w nvsga n [8] an [8]. Sufa an Lab wavs hav aso bn o fo suus o n on nson n [83] [87]. Fna bu wavs hough a fn-hnss a w nvsga n [88]..3. Ban gas h oa wav f a an gvn oson whn a honon sa s a suaon of h ans an f wavs whh a nf onsuv o suv as shown n Fgu.4. h nau of hs nfn s n b h wavngh av o h

37 ha. Sa Phonon sas n On Dnson 9 sang an fng faon. h hnonon of fqun ban gas n o suus s nown as Bagg fon fo h a [75] an sus fo hs nfn. Fgu.4. onsuv an suv nfn of ans an f wavs n a honon sa. No ha h wavs n quson a onguna an no ansvs. h aows a uv o a n vsuazng h has. h n ff of a h onsuv an suv nfn whn a honon sa s a anssson su ha onans ban gas whh ou whn wavs o no oaga whn a honon sa fo sf angs of fquns. Wavs of hs aua fquns a anua as h oaga hough a honon sa. Fgu.5 shows ha as o s a a o h honon sa h fquns whn a ban ga bo nasng anua. a π Fgu.5. Sua ban gas a of a D honon sa. As o s a a o a honon sa h ban ga anuaon nass.

38 ha. Sa Phonon sas n On Dnson h aous an ns h anssson an fon a a aa nfa as n (.) an (.) sv. As h nuson an nass av o h hos aa h ban ga wh an anuaon nas as shown n Fgu.6. a h ban ga wh s asu a h -3 B ng ossng ons an s noaz b h ga n fqun. In h au anssson gah of Fgu.6 osons o h -6 B ons. Anoh wa o ons hs snao of nasng an onas s o nvson h hos an nuson onssng of na aas n whh as h wou b no fon an a wav wou s ass hough unaff b h nfa. As h onas s nas h fon nass an nfn bos v-o oan. Evnua h nuson an s so hgh ha an b ons nfn an a of h wav ng s oa f off of h fs nuson. A sa snao hos f h hos an s nas wh s o h nuson an. In fa Fgu.6 wou b na f h gn a. anssson (B) a π Fgu.6. Ban ga haass fo vaous an aos. h sua sus fo fv s shows ha as h onas bwn h ans nass h ga wh an anuaon nass. h fng faon η an h has vo n h has hang of a wav as avss hough ah n. hus h hnsss of h vaous sons ha a u a an b hough of n s of wavnghs. Fo a h hnss of an nuson n s of nub of wavnghs s ηaf wh f s h fqun of ns. I s no susng fo hs assssn ha ausng h fng faon an h has vo has v sa

39 ha. Sa Phonon sas n On Dnson (bu nvs) ffs on h anssson su as usa n Fgu.7 an Fgu.8 sv. Dasng h fng faon us h av hsa hnss of h nusons an nasng h has vo us h nub of wavnghs ha f whn an nuson fo a gvn fqun. a π Fgu.7. h anssson sa fo vaous fng faons (η). Whn h honon sa s s ( η. 5 ) h anssson su s o. Howv as h fng faon s a nw ban gas a nou no h anssson su an h anuaon of h sng ons s hang. hough h nsonss fqun as us so fa s onvnn fo oang vaous honon sas s has nfoav o an h anssson su of a sa a wh a aas. Fgu.9 shows h anssson su fo onguna wavs n a fv- honon sa onssng of anang as of aunu (hos) an n (nuson). h sz a s an h fng faon s 35%. hus h aunu as a 3 h an h n as a 7 h. h aa aas a gvn n h fgu aon.

40 ha. Sa Phonon sas n On Dnson anssson (B) a π Fgu.8. h anssson sa fo vaous has vo aos. sua wo ffn aas hav ffng has vos an h g o whh h ff nasng affs h anssson su. h ff s sa o hangng h fng faon sn n h as h has shf bwn nfas s a. Fgu.9. h anssson su fo onguna wavs n an aunun honon sa. h sa onsss of fv s of an aunu hos wh n nusons. h aunu s 3 h an h n nusons a 7 h. h aa aas us fo h aunu a ρ 699 g an 63 an fo h n a ρ 89 g - an 563. s s

41 ha. Sa Phonon sas n On Dnson 3 unnng Fo a wav nn on a hn ba s ossb fo so of h ng o ass hough h ba. In hs on a ba fs o a sab of aa n whh h wav vo s u agna (.. an vansn wav); fo h anssson o b as asuab h ba shou b hn wh s o h wavngh. Fo a unnng an ou whn an aous wav s nn noa on a honon sa a fquns ns a ban ga an whn a wav s nn on aa a angs ga han h a ang a whh oa nna fon ous. Sn h wav s vansn whn h ba w hfo f off of h sufa f h ba s h. Howv f h ba s hn hn so of h vansn ng a ou o h u on h oh s of h ba an onnu o oaga. hus so of h ng a unn hough h ba. Fgu.. Iusaon of unnng hough a sng ba. A honon has boh a an wav os an has a oaon n b a saa obab sbuon. As suh whn s na a ona ba has a obab of sng on h oh s of h ba an hus has a obab of unnng. h hnn h ba h o s ha h honon w unn. In h fgu P f an P ans a h obabs of honon fon an anssson sv. I s ossb aso o vw unnng fo h sv of a sng quanu hana honon whh s a quanz o of vbaon an has a saa obab sbuon govn b h squa of s wavfunon. Jus as fo an on s no oa a an a on an hn a b sn oa as a fuzz sh wh a hgh ns n h n as n Fgu.. In hs as as a honon aoahs a hn ba s wav funon an na ns h ba so ha h s a fn obab ha h honon an b oa on h oh s of h ba. h hnn h ba h o ha h honon w b

42 ha. Sa Phonon sas n On Dnson 4 ans nsa of f. h fuzz sh s sa n h ans as o sn ha h obab of anssson s ow han ha of fon fo hs ba. Anoh unnng onfguaon s ossb usng a oub ba nsa of a sng ba. In hs as sanng wavs souons s bwn h bas shown fo h s as of bas of nfn hgh n Fgu.. (No ha fo a bas h hgh s fn a suaon ha s qu fo h wavs o na no h ba so ha non-zo anssson s ossb.) In hs su h sang bwn h wo bas nabs sonans o ou fo honons whos wavngh fs ns suh ha as usa n h fgu. Baus of h song sonan nhann ns h av fo b h wo bas h anssson hough h wo bas s % n h absn of an ssaon whn hs onon s sasf. Fo oh wavnghs h anssson s u no on baus of h uon u o unnng hough h wo bas bu aso baus hs honons w no f bwn h bas an a hfo wh nasng obab as h wavngh vas fo hos ha a a ah wh h ba sang. Phonon sas an a as honon bas a ban ga fquns wh aous wavs ungo suv nfn. Man sahs hav us hs o of o o h unnng of honons hough honon sas a ban ga fquns [6][89] [94]. Sng ba [89] [9] oub ba [9][93] an o oa wav ng shs [94] hav bn onsa na. n 4 n 3 n n Fgu.. Iusaon of os aow n an nfn oub ba. Whn a oub ba sos a a quanu w on sf wavnghs a aow. In aua a wav a s whn an nfn oub ba whn an ng u of haf of s wavngh s qua o h ba wh. On hs wavnghs an unn hough suh a ba.

43 ha. Sa Phonon sas n On Dnson 5.3. Dsson In f sa h s a na aonsh bwn saa an oa fquns fo an wavs anng ha a wav of a gvn fqun has a nown wavngh ha ns on h u n whh h wav s avng. hs na aonsh an b sa as f λ (.4) wh f s h fqun λ s h wavngh an s h has vo. Anav hs an b sa as (.5) wh s h angua fqun an s h wavnub. In a o aa h s no ong a na aonsh bwn an. hs an b onsa b onsng h na han of asss an sngs suss a. B assung ha h a sang n (.4) s a so ha ( na ) wav souon ( ) n hs gvs Sn osθ sn ( θ ) a a ( ) a an hn subsung a an (.6) a ( osa). (.7) a hs an b wn as sn ( a ). (.8) a hus w a fn h has vo as h ao of angua fqun o wavnub: ( ) φ. (.9) hs s a h sson aon of h ss an s o n Fgu.a). Pvous whn vng h wav quaon w ha assu ha h wavnghs of ns w uh ong han h sz of h s asss an sngs us n ou aoaon o h fu u. hs assuon as o h na sson o n Fgu.. Howv as h wavngh bos sa an aoahs h sa sz sa as h s asss an sngs n hs as h nvua ous an aos of h fu h sson bgns o va fo h ong wavngh assuon. h saa suu of h ss bos nasng van. hs sa ons a o honon sas: A ong wavnghs h sa aas o b a hoognous u wh ffv aa aas bu a sho wavnghs av o h sa sang h saa suu affs h sson.

44 ha. Sa Phonon sas n On Dnson 6 Fgu.. h sson aon a) an gou vo b) fo a s o ss. In h ong wavngh h sson ahs ha b a wav quaon n a onnuous u. Howv as h wavngh aoahs h sz of h o of h ss h sson s aff b h saa aangn. Anoh oan asu of sson s h gou vo whh an b fn as h a of hang of angua fqun wh s o wavnub ( ). Fo a suffn naowban us h gou vo an aso b ons as h oagaon vo of h sgna nvo. In a hoognous aa a ong wavnghs s s a onsan qua o h has vo sn. onsng h na han of asss an sngs g os( a ) (.) whh s shown n Fgu.b). A ong wavnghs h gou vo s onsan an qua o h f-sa onguna has vo. Fo sho wavnghs h gou vo aoahs zo osonng o a sanng wav. h aon of o Fo an nfn ong honon sa h f vaus whn on a suffn o n h f vaus a an oson n h sa. hs s a v sf wa of sang h Boh ho sos a h Boh-Foqu ho [95]. As a onsqun on wavnubs fo o π a a ons sn sho wavnghs (hgh saa fquns) a aas no hs ang b h o of h ss. hs an b nvson as h saa quvan of h Nqus Shannon sang ho.

45 ha. Sa Phonon sas n On Dnson 7 f (MHz) a π Fgu.3. h sson uvs fo an aunu-n honon sa. h bans a shown fo boh osv an ngav wavvos. Fqun ban gas ou wh h wavvo has a nonzo agna oonn nang ha h wav w a onna whn h sa. h aa an sa aas a na o hos us n Fgu.9. o usa hs on h sson uvs fo h sa aunu-n honon sa as n Fgu.9 s shown n Fgu.3. oason bwn hs wo fgus shows ha h ban gas n h fv- fn sa n Fgu.9 ou a h sa fquns wh h agna oonn of h wavvo fo h nfn sa shown n Fgu.3 s nonzo. hus a wav wh a fqun n on of h ban gas w b anua as oagas hough h sa an h oa anuaon w n on h san av an h agnu of h agna oonn a ha fqun. hs s wh a a fn sa w s ans so of h fquns n h fobn ban ga an wh ang o s fuh anuas h anssson whn h ban ga.

46 ha. Sa Phonon sas n On Dnson ouga ub wavgus Fgu.4. A ouga ub wavgu s a 3D suu ha bhavs as a D honon sa fo wavnghs ong han h ags ub a. A ouga ub wavgu s a h nsona suu ha as as a on nsona wavgu fo ong wavnghs an s usa n Fgu.4. A a an u-off fqun ansvs os ou an h suu no ong bhavs as a D wavgu [7]. A ossson of on of a ouga ub aous wavgu s usa n Fgu.5 wh a s h ngh of a sgn s h a of son an η s h fng faon. On onsss of a sng ougaon of ngh a a u of boh son an son. h souf wav quaon fo hs ub an b wn as u S ( ) S u ( ) wh S( ) S S ( η ) ( η ) < a (.) a < a u s h aous a san s h has vo n h oagaon u an S π 4 s h oss-sona aa of son. D oason of hs wav quaon wh h na D wav quaon as v a as o h fnon of ffv aa aas aua as ρ S an K S wh ρ an K a h ffv ns an bu ouus sv. hs ans ha a D honon sa a wh h sa sz an fng faon (a an η) an usng aas wh os ρ an K w bhav h sa as a osonng ouga ub wavgu wh oss-sona aas S n a u wh has vo. Fo hs ffv aas h aous an of ah son an b ss as ρ K S.

47 ha. Sa Phonon sas n On Dnson 9 Fgu.5. A sgn of a D o aous wavgu onanng wo sons. A ouga ub wavgu s an aav suu fo anass baus s a anaogous o a D honon sa n h sa as ovs so sfaons n h na as whh w b suss n ha 5. In h sa as h has vo n a ouga ub wavgu f wh a hoognous fu s na houghou h ub ( ) whas a honon sa a hav. Howv f anang ouga ub sons a f wh ffng fus hn h has vos a aso b ffn..3.4 Anoaous Do ffs h w-nown Do ff ous whn a sou v o fo a nonsaona. In a hoognous u wh a fo ha s ovng ona wh a sou an v h ans fqun w b u- o own-oua so ha wh f ( v ) f v < (.) f an f a h ans an v fquns sv an v s h vo of h fo. hs snao s n Fgu.6a). hs quaon an b unsoo b hnng of h fo as ovng hough aona wavnghs son whn ovng owas h sou an ovng hough fw wavnghs son whn ovng awa fo h sou.

48 ha. Sa Phonon sas n On Dnson 3 a) v b) v Fgu.6. gua a) an anoaous b) Do ffs n hoognous aas an honon sas. In a) h f wav fo a ovng fo whn a hoognous aa s u- o own-shf n fqun wh s o h nn wav nng on h vo of h fo an h has vo of h u. In h wo-oonn honon sa shown n b) h f fqun shf ns on whh u h fo s n a a aua an hus s a su of fquns nsa of a sng fqun n h hoognous as. h sou s shown as a s on h gh of h aa an on h nn wav s shown. Whn h aa s o as n a honon sa h has vo hangs oa as h fo ovs fo on u o anoh. hus h f fqun aso hangs oa an a f su sus. Fgu.6b) usas hs on. In [7] h auhos us a sgh ffn su ha onsa hs on wh a ouga ub wavgu sanng on-n. h a a ovng sou b ong an MP3 a wh a sa hough h ub. hough h has vo s onsan n h ub aaon of a ovng sou o h ouga ub wav quaon (.) sus n a v sa suaon. h sus onsa anoaous Do shfs ha aua ah h ons. In sonan aaas wh a ngav has vo h Do shf s vs an s a h nvs o vs Do ff. I s vn b onsng a onsan bu ngav has vo n (.) whh ffs fo h anoaous Do ffs n honon sas shown n Fgu.6b) wh h has vo ouas bwn wo

49 ha. Sa Phonon sas n On Dnson 3 osv vaus. h vs Do ff has bn o na n boh oagn [96] [99] an aous [] aaas..3.5 Aaons h aaons of D honon sas fo bu aous wavs (BAW) a os o fs [] [3] n aua fo soaon of ao fqun (F) an owav fs an sonaos n ounaon vs. Iagng an wavgung aaons a s o D an 3D suus sn wav faon os no ou n on nson. Bu aous wav fs a D honon sas sgn o hav a ban ga a unsab fquns. h an b faba unnah snsv F oonns on an nga u (I) o h soa h oonn fo vbaons. h has aso bn nvsgaon no on nsona sufa aous wav (SAW) honon vs [4] [7]. Sufa aous wav ansus ofn a h fo of nga zos os on a subsa ha sb a honon sa an h anssson os a o n [5]. h anssson of Lab wavs (fua an nsona wavs) hough D o suus has aso bn nvsga [6][7]. h hav aso bn nsng ns ha us aouso-o (AO) naons ha aow a SAW o of h os of o hoon vs (s fo a [8]).

50 ha 3 SAI PHONONI YSALS IN WO AND HEE DIMENSIONS h bas ons unng honon sa ffs n wo an h nsons a sb. L h D ounas h aso show ban gas an sson bu n aon h an aso hb anoaous faon whh offs h ona fo unqu fousng aabs. A of hs ffs an h unng auss a sb. Fna w sb h na sgn an haazaon of a D sa ha hbs ban gas an ngav faon. 3. Phonon sa ho an ffs h sa ns of aous wav sang ang o honon sa ffs as sn n on nson aso a o wo an h nsona sas. onsuv an suv nfn as o ban gas an h o of h ss auss sson. Howv sn w a no ong s o on nson a s ahaa son of hs ffs suh as ha gvn fo sson n Son.3. s usua no ossb n hgh nsons. hs ffs us b o usng a o oa ho h FDD o MS hos sb n a n ha Aous wavs n wo an h nsons Ahough h oagaon of aous wavs n wo an h nsons s sa o ha sb n Son. h a so noab ffns. On a s nvson a sf o of ffna vous of aa sn as asss an sngs as usa n Fgu 3. fo wo nsons. Longuna wavs a s suo n boh sos an fus. Howv h s now h ossb of a wav ha oagas wh sans nua o h on of oagaon a a sha o ansvs wav. Sha wavs w oaga n sos bu a no suo n nvs fus sn shang oons o no a fos o nghboung ffna vous of aa. hough sha wavs an an o ou n a D honon sas onssng of sos h a usua no ons whn h on of wav oagaon s noa o h aa nfas as was assu n ha

51 ha 3. Sa Phonon sas n wo an h Dnsons 33 Fgu 3.. Eas wav oagaon n wo nsons. h n ffna vou of aa s sa fo s qubu oson b an na fo na b a so aow. h oon of hs aa as fos o nghboung ffna vous of aa n boh onguna an ansvs ons. h onguna on s ona wh h on of wav oagaon an sus n a wav wh a ushng an ung ovn. h ansvs on s nua o h on of wav oagaon an sus n a wav wh a shang oon. Invs fus o no suo sha wavs bu sos o. h has aous wav an f o an hana san wav n an oagaon u. h has as wav s usua sv fo wavs n sos ha hav boh onguna an ansvs oonns. hs wavs anno b sb b a s saa vo ona whh an b sn b h foowng os of vo auus. h aous a san s gvn b h vo f u ( ) an a b wn as u ( ) u ( ) u ( ) wh u an (3.) u a h onguna an ansvs f oonns sv. h onguna oonn of h f sasfs h onon s u an h ansvs oonn sasfs u [66]. Sn h saa vo ona φ s fn as u φ hn h ansvs oonn us b zo sn h u of h gan of an saa f s zo ( ) φ. hfo a saa vo ona s nsuffn o sb a vo san f onanng ansvs wav oonns [9]. h sn of ansvs wavs n sos oas h anass of honon sas baus of an ff a wav o onvson. ons a D o 3D honon sa onssng of so sas n a fu. So of h ng of h onguna wavs ognang n h fu na o h honon sa nn on h sas a b onv no ansvs wavs whn h sas. Sa so of h ansvs wav ng ng a sa a b onv ba no onguna wavs. Fo hs asons h os an

52 ha 3. Sa Phonon sas n wo an h Dnsons 34 suaon hos us o anaz so-fu o so-so honon sas us aoun fo h vo nau of h aous wavs. Sang Aous wav sang n wo an h nsons s uh o oa han n on nson as was usa n h sa f fo wo soun-ha ns sn n Fgu.4. h oa f onsss of h nn an sa fs as shown n Fgu 3. whh was gna usng u sang ho. h sa f of fo a sng sa s hgh nn on h wavngh of h nn f an h ova anssson os of u sas a hgh nn on h sang bwn h nvua sas. Fgu 3.. Aous wav sang off an nfn ns n n wo nsons. h oa f onssng of h nn an sa fs s shown n a). h o of h sa f s vn n boh h fa-f b) an na-f ). h nn f ) wavngh s on h sa o of agnu as h sa a. 3.. Ban gas an wavgus h anssson su fo wo an h nsona honon sas w onan ban gas un an usans. h ou a fquns ha su n suv nfn bwn h nn an sa fs. On aona onsaon s ha h s a

53 ha 3. Sa Phonon sas n wo an h Dnsons 35 ona n o wav oagaon n hs sas ha was no sn n D an so a fqun ban ga a on b sn n aua oagaon ons. Whn h ban ga noasss a ossb ons of oagaon s a a o ban ga. A wavgu a b a n a honon sa a ban ga fquns ha wou ohws no oaga whn h sa. hs s ahv b ovng h sas fo on n of a sa o h oh aong a ah fo whh h wav w b gu. h wav w b onfn o oaga whn h sas of h ssng sas an w no b ab o n h honon sa. Anoh sa ff a oazaon ous whn sas a ov bu h s no nan o o h vo. Shou a wav b whn h vo w an saona o oaz an aas as a sanng wav. h has bn an onnus o b a vas aoun of sah anng o honon sas n wo [] [63] an h [5][6][89][64] [74] nsons. In wo nsons anass hav nu a s ow of ns [][] o o ohnsv sus of D honon sa os nung h ffs of sa sha an sz h a sha aa has (so o fu) an aa os on h ban ga wh an n fquns [] [9]. Oh sah has n hs gna onusons o nu h ffs of vsos [3] an agno-o-as aas ha hav oung of h agn an hana san fs [3]. Oh gous hav obn sabs of ffng honon sas o a v boaban fos [3][33]. An nsng a s a snso a b aowng hos n a honon sa a o f wh an unnown qu. h shf n s ban ga os hs o n h os of h unnown qu [34]. Bu an bnng wav os fo vaous honon sa as an sabs hav aso bn nsv nvsga [35] [4]. h aaon of D honon sa ns has bn n o Lab wavs [4] [47] an sufa aous wavs [48] [5]. As w h hav bn nuous suaons an onsaons of wavgung an oazaon usng D honon sas [53] [6]. In aua on gou sussfu onsa h us of a D honon sa wavgu as a fqun u [63]. h nsona honon sas hav aso bn su [5][6][89][64] [73]. In aua honon sas w a an w haaz n [5] an [6] an onsa an nsng ffs suh as ban gas an ngav faon. Sa sus of 3D honon sa os as n h D as hav bn fo nvsgang how h sgn aas aff h ban ga os [89][64] [7]. An nvsgaon of h sufa aous wavs sn on a 3D honon sa has aso bn fo [7]. On aua unqu aaon of h ban gas n honon sas was n h sgn of a sa wh a

54 ha 3. Sa Phonon sas n wo an h Dnsons 36 ban ga a ha honon fquns [73]. onsqun onsa v ow ha onuv. Wavgung hough 3D honon sas was o n [74] Dsson an faon Dsson ous whn h has vo ns on h oa fqun of a wav. Aous wav sson n wo an h nsona honon sas ass fo h saa o of h ss us as n h on nsona as sb n Son.3.. Howv h s now a ona oonn o boh h gou an has vos. As vous suss wavvos bon a aua agnu fo a gvn on a aas ba no sa wavvos baus of h o of h ss whh s a ban fong. Dsson an faon a na onn n wo an h nsona honon sas as w b usa. Wav faon ous whn h on of oagaon of a wav hangs u o a hang n h has vo. I s oan n honon sas baus of h unusua sson os whh an a o anoaous faon. o ons wav faon a h nfa bwn a noa hoognous u an a honon sa s fs oan o unsan wav faon bwn wo gua soo a. In a gua soo u wh has vo a onnuous an wav of a gvn fqun w hav a wavvo ˆ (3.) wh s h agnu of h wavvo an ˆ s a un vo n h on of oagaon. Whn h wav s nn uon an nfa wh a son aa havng a has vo h a a noa an a angna oonn of h wavvo wh s o h nfa. hus ˆ ˆ (3.3) wh an ˆ a h noa an angna oonns sv. Aso no ha ˆ sn h oonns a ohogona. o fn h wavvo whn h son aa h a us b a hs nfa: h angna oonns of h wavvo us b qua h agnu of h wavvo us oson o h has vo whn ah u an h on of gou vo noa o h nfa us b qua. Fo h fs on w hav h son on s. (3.4)

55 ha 3. Sa Phonon sas n wo an h Dnsons 37. (3.5) An fna h as on s sasf baus h gou an has vo a aa n boh a. obnng (3.4) an (3.5) an nong ha h noa oonn of h wavvo s. (3.6) h sus of hs ou fo wo aas s usa n Fgu 3.3 wh. 5. In h a θ was hosn o b 3º. No ha h angna oonns of h wavvos a qua an ha h agnu of has bn sa aoa. h sung ang θ s 9.5º whh s n agn wh Sn s aw.. 5 θ θ ˆ ˆ Fgu 3.3. h wavvos a h nfa bwn wo soo a. h oonns of h wavvos a noa an angna o h nfa an na as an sv. h angna oonns of h wavvos a qua as s qu fo h wav o b onnuous a h nfa. h u on h f has a has vo.5 s ha of h u on h gh whh aouns fo h nas wavvo on h gh. h a o aans on h gh baus of h sow has vo an hus has a ong wavvo. As a onsqun h ang of nn hangs fo θ 3º o θ 9.5º whh s n aoan wh Sn s aw. Anoh ans of usang wav faon a hs nfa s o us qufqun uvs as shown n Fgu 3.4. hs uvs a gna b ong fo a gvn n a ossb ons. Fo h hoognous a un onsaon h uvs a s sn h agnu of s no a funon of oagaon on. hs a no b h as n honon sas as w b usa. h aus of a s h agnu of fo a gvn n ha u. h bass vos wh s o h nfa a aso shown fo a. o n

56 ha 3. Sa Phonon sas n wo an h Dnsons 38 h wav faon fo h fs o son u h na wavvo s o. A n s awn ang h angna oonn of hs vo an wh nss h son uv nas h vau of. I s nowoh ha h a wo vaus of ha w sasf h angna ahng onon bu on of h os no ob ausa. Sn h gou vo s aa o h wavvo n gua a w us hoos h wavvo on h gh sn h on on h f wou na ha ng s sohow anang fo whn h son u. ˆ ˆ Fgu 3.4. h qufqun uvs us o aua faon a an nfa bwn wo soo a. h uvs a s baus h agnu of h wavvos a no funons of oagaon on. Fs h wo uvs a awn wh h uv fo son u havng a ag aus baus h a o aans fo a gvn oa fqun han n h fs u. h angna oonns of h wavvos a ah b awng a n (h ash n) ang h agnu of h angna oonn of. Wh hs n nss h son uv nos h oaon of. h a wo ossb hos an h no ho fo s shown wh an hough as os no ob ausa. h o ho s onng o h gh awa fo h nfa. In a o aa h qufqun uvs a no nssa s anng ha wavvos a hav ffn agnus n ffn ons fo a gvn. Aso h gou vos a no b ona wh h wavvo. Fo an n vsuazaon of hs os s ha n [75]. h gou vo s ( ) (3.7) g

57 ha 3. Sa Phonon sas n wo an h Dnsons 39 o o sa hs n wos s h gan of h sufa of as a funon of. Fo h hoognous as sb h gan s n h sa on as sn as h agnu of nass h aus of h qufqun nass an ongas n h sa on. As a ooa f h qufqun uv s no a h gou vo w no b ona wh. a) M b) M X X ) Longuna ansvs M X u Wav Vo Fgu 3.5. wo ss of qufqun an osonng sson uvs. Pans a) an b) show h fs Boun zon fo a fous D honon sa wh a squa a a fo usaon uoss (s Son 4.4. fo h onsuon of hs zon). Eah uv of a aua oou sns h s of aow wavvos fo a gvn oa fqun whos oou sa s shown on h f n ). Pan a) shows h uvs fo h fs onguna ban an b) shows h uvs fo h fs ansvs ban. h sson o n ) s a b avsng fo Γ o X o M an ba o Γ as shown b a ash aow n a) an a on h boo as n ). An a of h qufqun an sson uvs fo a fous wo-oonn squa a honon sa s shown n Fgu 3.5. h qufqun uvs n a) an b) oson o h fs onguna an ansvs bans sv shown n ). h oou of ah uv osons o h fqun oou ba n ). h sson o s a b M

58 ha 3. Sa Phonon sas n wo an h Dnsons 4 foowng h ah na b h aow n a) an awn on h boo as of ) fo a. No ha h uvs a s fo ow fquns an bo nasng ss ua as h fqun nass. As a onsqun h gou vo bos nasng ss ona wh h wavvo. In Fgu 3.4 f h gou vo was anaa o a as ha an oos sgn n h on noa o h nfa hn h anav ho fo wou hav bn h o ho osonng o ngav faon. hs suaon an ou whn h qufqun uvs ona owas h ogn wh nasng fqun. Jus suh a suaon s shown n Fgu 3.6 whh usas h aua sson uvs fo a honon sa a u of N os n an A hos. h sa s a squa a wh a fng faon of.5. Oun n s a ban whos so s asng as h agnu of h wavvo nass ang o ngav faon Hz Ban a M X u Wav Vo Fgu 3.6. h sson uvs fo a D honon sa. h sa s oos of a squa a of N os n an A hos sa aa an havng a fng faon of 5%. h s a o 4 Hz ban ga sn bwn h fs an son gous of bans. An aa ha sas ngav faon s oun wh a ash ova an s nfab baus h wavvo agnu ass as h fqun nass. W hav no us h ngav fav n n h on of honon sas baus ha aua vau as o hoognous o ffv hoognous a. Insa w s f o ngav faon. On us nsu ha ngav faon s aua oung an ha s no us an afa of h o of h ss. An n susson abou h anng of ngav faon n hoon sas s gvn n [75] an h susson aso as o honon sas. Aso on us b afu o ons h o of M

59 ha 3. Sa Phonon sas n wo an h Dnsons 4 ffaon n ngav faon ns sn h ffaon an of a s fo aon w onbu sgnfan o h aaan of fousng [76] Fgu 3.7. Wav faon un noa an ngav fang onons. A noa nfa s shown n a) an a ngav fang (N) nfa s shown n b). h has s na b h wavvo an h ng fow s na b h ash vo S. h ansvs oonn of h wavvo a h nfa s aso shown. h aona onon ha h has an ng a anaa n an N aa as o h ngav faon obsv a hs nfas. h unusua suaon of ngav faon s usa n Fgu 3.7 whh shows wav faon a a noa nfa n a) an a a noa o ngav fang nfa n b). hus un gua aas a ana sab of ngav fav aa an b us o fous a on sou ono an ag an as n Fgu 3.8. Whas gua nss qu a uv sufa o aosh fousng ngav faon as an nna an na fous wh a s sab. Fgu 3.8. An usaon of fousng n an N sab. In a sab of noa aa shown n a) as fo a on sou onnu o vg bu a a ffn a whn h aa as n b h ao of has vos. In a sab of N aa h as ungo ngav faon uon nng an ng h aa ang an nna an an na fous (af Pn []). h sson of wavs oagang whn honon sas has bn nsv su n wo [9][77] [86] an h [5][6][89][77] nsons. Ngav faon has bn

60 ha 3. Sa Phonon sas n wo an h Dnsons 4 hoa an na o n D sas [9][8][77] [8]. Hgh ona wav o oung has aosh b hoosng a sou fqun v os o a ban g wh on on aow wavvo [83]. A unqu -vsa sh fo wav fousng wh a honon sa has bn sn [84]. Insng h ffv au of h honon sa s ag han h ngh of s sufa ossb baus of h nas a ah nghs u o u sang. h sson os of a honon sa wh zo sas ha aouns fo h E-f oung hav bn nvsga [85]. A honon sa wh saa vang sgn aas has vn bn shown o hb h sa wavgung os as a gan n ns [86]. h sson an fousng os of h nsona sas hav aso bn nvsga [5][6][89][77]. 3. Phonon sa sgn an na sus W sgn faba an haaz a wo nsona honon sa. h an uos of hs s was o onsa so of h nsng honon sa ffs nung ban gas an ngav faon. I aso sv as a ans of vfng ha h honon sa suaos sb n ha 4 ah h na sus. Aso an of h onsaons an ffus noun n h oss of ang an haazng a honon sa hav bn una hough hs s. h anssson an sson os of h sa hav bn haaz n h sa ann as sb n [87]. 3.. sa sgn h w a nub of onsans ha h o n h sgn aas of h honon sa. Fs A D suu was hosn baus a D sa wou no sa wav faon an a 3D suu was oo ffu o faba n-hous. W wan o us h subson an asung aaaus n ou ab so wa was hosn as h hos aa. hs aso sf h anass baus w no n o asu ansvs wavs sn fus o no suo ansvs wavs [66]. A na sa sha was hosn baus of s s. h anng sgn aas nu: Ds fquns of oaon Sa aas Fng faon sa aangn Nub of as Oh ana onns

61 ha 3. Sa Phonon sas n wo an h Dnsons 43 Fqun of oaon h s ang of fquns fo whh h honon sa w hb nsng ffs ns n a h sang an ova sa sz. hoosng a ow fqun ang ans ha h sa w b as o anufau an ha anufaung oans w b sa oa o h wavnghs of ns. Howv h sz of h oang nvonn s h au sz. In hs as h subson an s h oang nvonn. h s fqun ang was a s b h avaab ansus whh nu 5 Hz 5 Hz MHz an sva oh son ansus of hgh fquns. hs h ss of ansu as w haaz b ang h n h subson an n a hough-anssson onfguaon [66]. A Panas-ND o 58 us/v was us o gna an v an us. A on DS 3B ososo was us o au h -oan anssson wavfos whh w hn os-oss n MALAB. h v wavfos w avag ov an ansssons o u h ffs of nos. - Noaz Au (B) MHz 5 Hz 5 Hz Fqun (MHz) Fgu 3.9. h fqun sons fo h ss of ansu as. h ansus w a n as a hough-anssson onfguaon aoa 3 aa. h sus of h ansu haazaon a shown n Fgu 3.9. h MHz an 5 Hz ansus sa n haass whas h 5 Hz ansus w

62 ha 3. Sa Phonon sas n wo an h Dnsons 44 v nos. I was a n ha h 5 Hz ansu ha wo ssus: h w a sgn o ou o ansvs wavs n sos an on of h ansus ha a a on s sufa. Fo hs asons h 5 Hz ansus w no ong us. h ash n n h fgu sns h -6 B on an fo hs w an n h ansu os suaz n ab 3.. hfo h s ang of fquns fo h honon sa wou b whn h MHz an 5 Hz ansu sons uvs. ab 3.. ansu a aas. Paa MHz ansu 5 Hz ansu B Fqun 4 Hz 456 Hz Man n Fqun 64 Hz 47 Hz Low -6 B Fqun 86 Hz 8 Hz -6 B Fqun 3 Hz 66 Hz Banwh 46.6% 9.6% Au Da Sa aa As sb n ha h sa sz an an n a honon sa aaa affs h sa os. h sa aa ho s oan o a ss g. h os sgnfan fao n hoosng h sa aa s ha s aous an an has vo b hgh onasng wh h hos aa. Sn w hos wa as ou hos aa an sff so wou hav asa ffn os. I was on ha w us sanss s ao 33 (abbva as sanss 33 ) baus of s ab o ss ooson n wa. ab 3. ss h aa os of h hos an sa aas. h sffnsss gvn oson o h Vog noaon fo an soo aa [75]. h has vo an haas aous an w aua fo h ns an bu ouus. Fng faon W o us a fng faon of η. 58 bas on h D sas sn an haaz n [9][74][77]. hs fng faon was hosn o ahv h au ban ga wh fo a D squa a.

63 ha 3. Sa Phonon sas n wo an h Dnsons 45 ab 3.. Hos an sa os. Paa Hos Sa Na Wa Sanss 33 Dns ( ρ ) 998 g g -3 ( ). Pa 3.6 Pa Sffnss ( ). Pa 77. Pa ( 44 ) Pa 77. Pa haas Aous Ian ( ).48 Ma 43. Ma Longuna Vo ( ) 485 s s - Sha Vo ( s ) N/A 3 s - sa aangn W hos o us a s squa aa as s w-sabsh ha hs suu hbs ban gas an ngav faon as suss n Sons 3.. an 3..3 an h fns hn. Boh h ΓX an ΓM ons w onsa ban gas an ngav faon f h sa s o sgn so w hos o faba h sa n h ΓM on. As fo a sang w o s hs as a funon of h sa aus an s fng faon so ha a (3.8) η wh a s h a sang s h n aus an η s h fng faon. W hos o us /8 h π nh a o (a oan ±.5 nh) whh was n o sona n h ang aab o ou avaab ansus sung n.575. hus h a sang was a h avaab anufaung oss ha a oan of.54. hs vaus a suaz n ab 3.3. Nub of as Ia a honon sa wou b of nfn n n a ons whh s ofn an assuon a n h hoa anass. Howv h a of fabang a honon sa s ha a fn nub of ows an ouns us b hosn. H w an ows an ouns o b h nub of ans of ns aa an nua o h on of anssson sv. h nub of ows affs h g of anuaon whn

64 ha 3. Sa Phonon sas n wo an h Dnsons 46 ban gas as was usa n Fgu.5 fo on nson. h nub of ouns affs h g o whh g ffs an ffaon a a o n h v sgnas. o nz hs ffs h nub of ouns n h sa shou b az. Aso h FDD suaons of h sa assu ha was nfn ang n h oun on bu wh a fn nub of ows. Anoh onsaon s h an os assoa wh ahnng huns of ns ou of ong sgns of o. Wghng a hs faos was n ha 37 ouns an ows fo a oa of 4 ns wou ahv a goo baan bwn hs ong quns. ab 3.3. sa a aas. Paa sang (a) Sa aus () Sf Vau Fng Faon (η).58 La La Don Squa ΓM Oh ana onns In aon o h bas sgn aas of h sa h a oh faos ha n o b ons. hs nu h n ounng suu an h n ngh. Sn h au of h 5 Hz ansu was h ags wh a a of 8.6 was ha 5 ong ns w suffn o avo ssv fons fo h ounng suu.

65 ha 3. Sa Phonon sas n wo an h Dnsons 47 o sgn Fgu 3.. o vw of h honon sa sgn. h osss oa n h ns of h sa oaons a gus. A o vw of h D honon sa sgn s usa n Fgu 3. wh h os oan nsons ab. h osss sn whn h sa oaons a gus an a o a h fabaon oss. h fnsh faba sa s shown n Fgu 3.. No ha h s an aona gon of a ung ou n h o a of Fgu 3.a) ha s no sn n Fgu 3.. hs a aa was us as a oaon fo a ounng ho no whh a ha o was a o ho h sa n oson n h subson an. Fgu 3.. o a) an s b) vw of h faba honon sa. h a fv ssng ns on h n of h sa whh ou baus of aa oss ung h fabaon of h ns. I was ha h sn wou no sgnfan aff h anssson o sson os of h honon sa.

66 ha 3. Sa Phonon sas n wo an h Dnsons Ena hos Fgu 3.. Ena onfguaon. In a) boh h ans () an v () ansus a saona. hs was us fo obanng h anssson su an sson uvs. In b) h v ansu s a wh a hohon ha s as-sann n h an of h ns. hs onfguaon was us o ag h aous f. wo na onfguaons w us o haaz h sa. h fs n Fgu 3.a) was us o gan h anssson su an sson os of h sa. I us na 5 Hz o MHz ansus as boh h ans an v n an h w oa aoa 5 awa fo h sa fas. h oh onfguaon s shown n Fgu 3.b). I s na o h vous onfguaon ha h v ansu s a wh a hohon ha was as-sann n h an of h ns. In boh ass h ansus an sa w f saa whn h an b suong os sus h anssson su sng h onfguaon n Fgu 3.a) a fn us was sn no h an an o whou h sa. Wh h sa n a h ans us was o. h su of h v sgna noaz b h su of h fn sgna vas h anssson su of h sa an hs s shown n Fgu 3.3. h sua su s h su of an FDD suaon of a boaban us oagang hough h sa. oo agn bwn h a faus of h wo sa an b sn ov h ang fo o 8 Hz. Muh bon hs h wavngh aoahs h sz of anufaung fs an h sa aas o b nasng so whh a aoun fo h sans. In aon a no san of h ban ga gs bwn h wo sa shou b no.

67 ha 3. Sa Phonon sas n wo an h Dnsons 49 anssson (B) Fgu 3.3. Masu an sua anssson su of h honon sa. h asu su s oos of h wo sa oban fo usng boh h 5 Hz an MHz ansus an ng h sus ogh. h sua su was a fo an FDD suaon of a boaban us oagang hough h sa. h sson os h sson os of h honon sa an b n b anng h has shf bwn h fn an sa sa. hs uss h sa aa oban fo h anssson su n. sng h has shf of h asu sgna h agnu of h wavvo as a funon of angua fqun s wh φ( ) ( ) φ ( ) (3.9) L s h has ffn bwn h sa a angua fqun an L s h ah ngh fo on fa of h sa o h oh. Vaus of ga han π a a fo ba no h fs Boun zon as n [87]. h asu an sua sson os of h sa a shown n Fgu 3.4. Sn h sa was a n h ΓM on h asu sson uv os no gv nfoaon abou sson n h ΓX o XM ons. h FDD suaon ous a su fo v a wavvo an h as of ha su a na as aons n h fgu. Howv so as a of a ga agnu han ohs an sn h oagaon os ha a ng. hs a oun n bu fo h os bow 6 Hz. h sson uvs aua fo h anssson su usng (3.9) a shown n ash gn fo h fs wo bans an ban ga. I ahs w wh h sson. A oon of h anssson su of Fgu 3.3 s shown on s s on h gh of h sson uvs o a n nfng h ban gas. h fs h ban gas a na wh hash as an a agn o h FDD sus.

68 ha 3. Sa Phonon sas n wo an h Dnsons Fqun (Hz) Fgu 3.4. Masu an sua sson uvs of h honon sa. h sua sson uvs w a fo an FDD suaon of h sa fo vaous a wavvos. h as of h fqun sons fo ah wavvo a o as aons. h songs oagaon os fo o 6 Hz a oun n bu. h sson aua fo h asu anssson su of h sa s shown n gn fo h fs wo bans an ban ga. h anssson sa a a on h gh han s of h fgu o a n vsua oason. h fs h ban gas agn o h FDD suaon a na wh hash as. h so of h wavvo an b sn o b asng wh s o fqun n h son anssson ban. I s n hs fqun gon ha ngav faon shou b obsv. h aous f sng h onfguaon n Fgu 3.b) h aous f n h an of h ns was asu usng a ban hohon. A on AW5 us onn o a P-4 ow af was us o ans an uasoun us. h ow af was us o nas h ans sgna au o ahv an aab SN fo h hohon us. h hohon whh ha a 4 µ a snsv gon was a Sonoa Ma Sss 84 ban hohon. I was onn o a Panas-ND o

69 ha 3. Sa Phonon sas n wo an h Dnsons a an hn a Naona Insuns 5 gz. A boaban us was sn wh h sa n a an was o a gua sa g ons as a n ab 3.4. h an z ons a aa an nua o h fa of h sa sv. ab 3.4. o aous f aas Paa Vau Nub of wavfos avag oaon Pons wavfo (N ) 8 s ( ) Wavfo uaon ( a ) En n h on souon n h on ( ) 5 ns s 8 oa nub of ons n h on (N ) 8 En n h z on souon n h z on ( z ) 6 oa nub of ons n h on (N z ) 6 oa nub of saa oaons sa 494 Dsan fo ansu o sa Dsan fo sa o g of san gon SN B h v sgnas w hn onv o h fqun oan an a oou-o ag of h o saa f fo ah fqun was gna. nfouna h ow af nou a aas sgna no h ans wavfo u o h swhng on an off of a ga sgna. h ga sgna was qu o nab an sab h ow af ouu an ou no s b f on. I us b swh on fo ah ans us hn swh off afwa o h u of h af. I was hs swhng aon ha a h aas sgna shown n n Fgu 3.5. Ia shou b zo bu h ans us shown n bu a onans ns of h aas ga sgna.

70 ha 3. Sa Phonon sas n wo an h Dnsons 5 Noaz Au (B) Fgu 3.5. h oa an ga-on sgnas. h oa ans sgna nus boh h s nu us an h unwan ga sgna. A -oan wnow was us o soa h s sgna an o h aas ga sgna. On sag o na h ga sgna was o s suba fo h oa v sgna. nfouna sgh has os n h wo ss of sgnas a hs aoah unsussfu. h sag w o o us was o f h v sgnas n h -oan wh a Kas-Bss wnow n abou h sgna a. hs sussfu h ga sgna. Bas on h sson os of h sa w o fn ngav faon oung sgh abov 4 Hz na h o of h son anssson ban. o nf hs w a o obsv fousng of h aous f na o h sa. h na onfguaon s shown n Fgu 3.6a). h o bwn h ansu an h sa nas h naua fous of h ansu oa 68 fo h ansu sufa. h ash ns anang fo h fous usa how ngav faon as o h gnaon of a foa gon n h san aa. Fgu 3.6b) shows h aous f a 4 Hz an a foa gon an a b sn onsang ngav faon. h foa gon s sagn n h ag baus of offs os n h na su. hs fousng ff was on aan fo 49 Hz o 48 Hz an Fgu 3.6) shows an a of hs banwh. h f a 36 Hz os no show h fousng sn n h abov an. A o

71 ha 3. Sa Phonon sas n wo an h Dnsons 53 fnv oof of ngav faon wou b o a a wg s an obsv h ba angs ng h wg; howv hs was no a ossb onfguaon gvn h sgn of ou sa. 8 6 Fgu 3.6. A aga of h na onfguaon a) an h asu aous f a 4 Hz b) an 36 Hz ). h na onfguaon s shown n a) (no o sa). h san aa s shf o h gh of h nn baus of na onfguaon offs os. In b) an ) h ag oou sns h ogah of h noaz agnu of h ssu wh s o h f ssu au whn no sa s sn Sua W sussfu sgn faba an haaz a wo nsona honon sa. Ou sa onsa ban gas a h fquns b ou suaons hus hng o onf ha ou suaos w o n. h sa aso onsa sson ha o ngav faon. h boaban aous f was asu an oos no s oa fqun oonns. Ngav faon ang o fousng of aous wavs a 4 Hz was onsa.

72 ha 4 SIMLAION MEHODS In hs ha h bs sabsh os fo suang h bhavou of o aous sss a sn: a assu ha h a s ossss. In aua w sb h fnffn -oan (FDD) anssson a ho (MM) u sang ho (MS) an an-wav anson (PWE) aoahs. Eah of hs hos was n n MALAB an as a ans of ong boh hoa sss an h na honon sa sb n ha 3. In aua a ga a of wo wn no nng h FDD suao sn was h bnha agans whh ou nw vang hos w oa. h MM an MS hos fo h bass of ou nw vang os an h PWE ho was a us o gna qufqun uvs fo ou D honon sa. Fna so anav hos a bf sn aong wh a sho susson gang h anab of vaous hos o nooang -vang aa aas. In wo an h nsons h vo nau of wavs n so-so o so-fu honon sas us b aoun fo. h FDD an PWE hos aoun fo ansvs wavs bu h MS ho as sn os no. I an b of o aoun fo h vo nau of h wavs b nung aona f vaus an bouna onons bu h ns an h sa. Fo h sa of a w s h MS ho o onsng saa vo onas. 4. h fn-ffn -oan ho h FDD ho nua suas h aa ffna quaons ha govn h oagaon of wavs whn a oagaon u. h FDD ho has an bnfs: h sung ffn quaons a anab o sofwa naon; h sus o wha wou b obsv n a aboao snao; anssson an fon offns a as n fo h sus; ong a va of suus s ossb; an as nu wavfo an b us. So of h awbas a as foows: h s no os-fo ana souon o ua h suaon sus; h a sab a ha ana g souon s whh an sgnfan nas h suaon un. hus w hav us h

73 ha 4. Suaon Mhos 55 nua sus fo h FDD ho o vf h sus fo ou nw -vang hos. Po aous suus an b nvsga wh h FDD ho b suang on of h suu an nng o bouna onons. hs bouna onons ansa h ougong wavs a on bouna o nong wavs a h oos bouna wh an aoa has shf. h n su aas o b an nfn sa a u of na s wh an a wavvo. h oan san of a ano hosn on on h suaon g s o fo h uaon of h suaon. Afwas a Fou ansfo of h sgna w va whh wav fquns oson wh h os wavvo. h sson aon of h honon sa an b o b ang hs oss fo an ffn wavvos. Non-o suus a sua b ang a zo-wavvo ( ) o h sa s quaons as n h o as. wo s of bouna onons a us: fng an absobng. h fng bouna onons fo h bouna san o zo whh n fs nn wavs. h absobng onons a an aoaon o suang a gon of nfn n anng ha ougong wavs of a saa an oa fquns w h suaon bounas whou an of h ng bng f. hs az suaon s no n ossb an nuous shs s fo aoang hs onon. hs h bouna onons (o fng an absobng) aow h suaon of aboao- sus so ha na aa a b oa wh FDD suaons. h fb of h FDD ho s vn b s vasvnss n boh aouss an oagns. I has aso bn a o h f of ssoog o su os of wav oagaon hough h ah [88][89]. I has aa bn nsv us o su honon an hoon sas (s fo a [9][9]). I s v anab o o aazaon [69] whh s aua van onsng h n owas aa ouaon an u-o Ps. h ho has bn of o ov s aua b usng hgh-o quaons [9] ausng h sha of h unng s g [93] ovng oab wh absobng bouna quaons [94] an aounng fo ssv a [95]. W a an FDD suao n h anguag ha uss MALAB as a fon-n fo ovng aa o an anazng aa fo h suao. hs onfguaon aows h vsa aa anass of MALAB wh h s of uon of o o. Fuh foan gans a ossb b wng h -o o un on a gahs osso bu ha o s

74 ha 4. Suaon Mhos 56 ous h so of hs hss. W wo h suao baus h oa avaab aous FDD suaos w no aab of suang -vang aa aas. 4.. Dvaon of a D agoh H w v h sz govnng quaons fo h no gon of an FDD suao. W hn a vaous bouna onons o ffn usans nung o fng an absobng bounas. Po s assu n h vaon bu h n su s a s of s- quaons ha aso wo fo non-o suaons. h foowng onvnons w b us nhangab n h vaon whh a us o an onssn wh [35]: ( ) ( ) (4.) ( ) ( ) ( u u ) ( ) (4.) u u u (4.3) wh s h asan oona vo s h wavvo an u s h Lagangan aous a san fo qubu. Fuho w ons on ossss soo a wh a saa ns an ossb. W hoos o sov fo h vo san f sn hs w h suaon of ansvs wavs n aon o onguna wavs as oos o a saa f suaon whh an on nooa onguna wavs [66]. h a wo ss of quaons qu o sov fo h san f as was h as n Son.. h fs s Nwon s son aw of oon an h son s h sss-san aonsh of h aa. Fo a v a an nsv vaon sa o h foowng h a s f o. 4 of [75]. W bgn wh h ns vson of Nwon s son aw ρ u&& σ (4.4) wh σ s h sss nso ρ s h ass ns an u s h vo san. hs an b wn as ( ) σ ( ) u n ρ ( ). (4.5) h o of h ss s hn nfo anng ha h no suaon gon s assu o b nfn n boh h an ons whh s h aa naon Boh-o onon fo a gvn wavvo [95]. h Boh onons on h sans an ssss a u ( ) ( ) n n (4.6)

75 ha 4. Suaon Mhos 57 an ( ) S ( ) σ. (4.7) n n Subsung ba no h ogna quaon gvs ( ) S ( ) ( ) n ρ. (4.8) Eanng h vav an anng h oon onna ( ) n n ( ) ( ) ( ) Sn ρ n Sn (4.9) n n ( ) ( ) ( ) ( ) S n ρ nsn. (4.) n n Eanng h suaon ov wo nsons an hangng h noaon o ah [35] gvs h fs wo quaons n onnuous fo: an : : : : S : S ρ S S (4.) : : : : S : S ρ S S. (4.) o sz (4.) h foowng s aoaons o h onnuous vavs a us. h a hosn o n h fn ffns whn obn wh h son s of quaons fo S. S [96] fo o nfoaon on na fn ffns. Aso an a an o b na ns n h s as anng ha of hos aas an wh s on onnuous s s h sa o souon. hs onvnon s us o sf oason of h onnuous an s quaons. h s aoaons a : : : : (4.3) S : : : S S : : : S S S (4.4) an S : : : S S : : : S S S. (4.5) Ang hs aoaons o (4.) gvs ρ : : : : : : : K S K S K S K S (4.6)

76 ha 4. Suaon Mhos 58 wh aangng o sov fo ρ K ± ±. (4.7) : : : : : : ( K S K S K S K S ) :. (4.8) Equaon (4.) a aso b sz usng h foowng s aoaons an b nong ha S S whh s a onsqun of h ss sb subsqun: : : : : (4.9) S : S : S : : : : S S S (4.) S : S : S : : : : S S S. (4.) Ang hs aoaons o (4.) an aangng gvs ρ : : : : : : K S K S K S K ρ S : : : : : : : ( K S K S K S K S ) : (4.). (4.3) h son s of quaons qu o sov fo h san f s h sss-san aonsh whh s ou b h oson-nn as sffnss nso so ha Enfong h ss o as bfo ( ) u σ ( ) n ( ). (4.4) S n ( ) ( ) n n n n ( ) (4.5) Eanng h vav anng h oon onna an nong ha s an n as w as S ( ) ( ) ( ) S n n ( ) ( ) ( ) n ( ) ( ) n (4.6) n n. (4.7) n n

77 ha 4. Suaon Mhos 59 As sb n [75] an of h sffnss nso s a unan u o ng onsvaon us. Foowng h Vog noaon (suaz on. 4 of [75]) h van s a shown onns n h a. Assung soo os (a on fn n [66]) whh ub s us h offns u o hos shown n h a on h gh (. 9 of [75]): Vog Noaon ub S. (4.8) Eanng h suaon n (4.7) ov wo nsons an hangng h noaon o ah [35] gvs h onnuous fo of h sss oonns: an : : : : : S (4.9) : : : : : S (4.3) : : : : : : S S (4.3) As bfo hs quaons a sz usng s aoaons. h sung s quaons aus o ahv na ffns a : : : : : S ( K K ) ( K K ) : : : : : S ( K K ) ( K K ) : an ( : : : S K K K K : ) (4.3) (4.33). (4.34) 44 hus fv s quaons ((4.8) (4.3) (4.3) (4.34) an (4.33)) an fv unnown sa vaabs ( S S an S suaon. Fo a o suaon wh a gvn wavvo ) fu sf h no gon of h FDD K n (4.7) s o an hus h no gon sa vaabs a o. Fo a non-o suaon an h sa vaabs a a. 4.. Inaon Offs g h s quaons a aua usng an offs g as shown n Fgu 4.. hs aows na ffns o b us n h aoa vavs whh sus n ov

78 ha 4. Suaon Mhos 6 aua of h suaon [96][97]. h aa aas a aso oa off-g as shown n h gh-han oon of h fgu whh us b aoun fo whn onsng v fn suus n aon o h g sang. Fgu 4.. h offs g naon of h FDD sa vaabs. On h asan g shown o h f h ba os na vaabs oa on-g. ha s h a oa a an n wh an n a ngs. h wh os na vaabs oa off-g n h h o boh as. ha s an/o n a ngs ±. hs aangn s h us of na ffns us n h s aoaons. h s quaons fo h FDD sa vaabs a now b aus o sb h offs g us n Fgu 4. whh s h foowng s of quaons an a h sa as sn n h An of [35]. No ha h quaons n [35] fo ± K a ssng an (o usng h noaon n hs hss) an o ha was onf wh h auhos. ( ) : : : : : : : S K S K S K S K ρ (4.35) ( ) : : : : : : : S K S K S K S K ρ (4.36) ( ) ( ) K K K K S : : : : : (4.37) ( ) ( ) K K K K S : : : : : (4.38) ( ) K K K K S : : : : 44 : (4.39) h offs g s of no aa onsqun n h aua sofwa naon. I on qus ha h us s awa ha h aa aa nus an h f vau ouus oson o ons oa on h offs g.

79 ha 4. Suaon Mhos 6 Bouna onons In an FDD suaon usng h s quaons abov h no gon a b aua n fo ov na onons an aa aas. Fo h suaon o b anngfu wo oh s of nfoaon us b ov whh a h sou an bouna onons. Wh s o bouna onons h a h usfu s ha w n n h FDD suao: Po fng an absobng. Po bouna onons Po bouna onons a h f vaus fo on n of h suaon gon an a h o h oh wh a has shf n b h a wavvo. h ason fo usng o bounas n a suaon s o a h no sua gon aa o a nfn n h nson of h o bounas. Fo a a honon sa an b a o aa nfn n a ons b sng a fou bounas o b o. h sson aon of suh an nfn sa an b sov b ang a wavvo o h suaon an asung h fqun su of a on n h no. As an a of a us of o onons n on on nson h sa ha was onsu an sb n a n Son 3. was sua b assung ha an nfn wh an a fn h whn n a ha 37 ouns of wh an ows of h. h sua an asu anssson su ah qu w as was shown n Fgu 3.3. In ah san an sss f h s nough nfoaon o aua wo of h bounas an h oh wo us b o usng o bouna onons. ab 4. suazs h gons ha an b ou an h bounas ha us b o. In h ab X an Y sn h oa an sans bng sua an M an N sn h au an ns sv so ha X M an Y N. h bounas a o fo h oos s of h suaon gon wh a has shf ha ns on an h sz of h suaon gon. h unsf on vaus an b f unfn sn h o no nfun an oh s.

80 ha 4. Suaon Mhos 6 ab 4.. auab suaon gons an o bounas. F qu gon auab Ins Bounas Bouna onons M N X M Y N M N M X N Y S /S M N M X S / S/ S / Y N S/ S M N X M S S N Y S S h fow of h FDD agoh wh o bounas s n Fgu 4.. Fgu 4.. FDD agoh fow fo o bounas. h no sss fs S a aua hn o bouna onons a a. N h no san fs a aua an h o bouna onons a a. Fna h s s nn an h oss s a.

81 ha 4. Suaon Mhos 63 fng bouna onons A bouna an b a fv s b sng a h f vaus o zo aong h bouna. hs aso wos o a fos n h no f qu. I s a onvnn wa o sua a hgh onasng aa suh as an aunu bo n wa. Absobng bouna onons h ob of ang a wav oaga ou of an FDD suaon gon whou an ng fng ba n s no va an an shs fo ahvng hs hav bn oos [98] [3]. I qus h sn of a f ah a (PML) ha s non-fv fo h saa an oa fquns of ns an ha an aso ssa h nn ng. h ob s sowha sf whn onsng onohoa sous ha a noa nn. Howv ougong wavs ha a nasng aa o h bouna bo v o f ba no h no suaon gon. A nw PML absobng a has bn oos ha asss an of h shoong of vous os [4]. I has bn shown o b f ah an sab wh o aa hos [5] an has vn bn sussfu a o h nonna wav quaon [6]. Fo hs asons hs aua PML o was us n h FDD suao fo absobng bouna onons. I s wohwh o sa h PML govnng quaons n boh onnuous an s fo sn h a gvn n [4] as onnuous an nsonss whas h FDD suao qus s an o nson quaons. h 3 nvua govnng quaons a gvn as h a quaons n whh a nos ffnaon wh s o h vaab. h h a quaons a ( E ) ( v ( ) E w ) B v ( ) v A w fo (4.4) ( ( ) α ) w w E v (4.4) an w E v ( ( ) α ) w. (4.4) In hs quaons h sa vaabs a o hos n h no suaon gon a onan n v an h PML sa vaabs a nou n w an w. h a u u v σ σ σ 3 h as of h offns a w w w w w 3 4 an w w w. (4.43) w 3 w4

82 ha 4. Suaon Mhos ρ ρ A 44 ρ ρ B (4.44) E an E. (4.45) h ang funons ( ) an ( ) a osv an sooh an a zo a h bouna. h hav bn s o a hbo angn funon whh nass n vau fo zo a h bouna o a a h fa g of h PML. h aas α an α h o vn gowng osaons as h suaon ogsss [4]. hs govnng quaons a an n ab 4. an a h bass fo h s quaons. ab 4.. onnuous PML absobng bouna quaons. onnuous PML quaon (4.46a) ( ) ( ) w w u σ σ ρ (4.46b) ( ) ( ) w w u σ σ ρ (4.46) ( ) ( ) w u w u σ (4.46) ( ) ( ) w u w u σ (4.46) ( ) ( ) 44 3 w u w u σ (4.46f) ( ) ( ) w u w α (4.46g) ( ) ( ) w u w α

83 ha 4. Suaon Mhos 65 onnuous PML quaon (4.46h) ( ) ( ) 3 3 w w α σ (4.46) ( ) ( ) w w α σ (4.46) ( ) ( ) w u w α (4.46) ( ) ( ) w u w α (4.46) ( ) ( ) 3 3 w w α σ (4.46) ( ) ( ) w w α σ h onnuous quaons a sz usng na ffns. h s quaons a suaz n ab 4.3 wh h suffs of h onnuous an s quaons ah. ab 4.3. Ds PML absobng bouna quaons. Ds PML quaon (4.47a) ( ) ( ) : : : 4 : 3 : 3 : 3 : : : u u w w u σ σ σ σ ρ (4.47b) ( ) ( ) : : : 3 : : : 4 : 3 : 3 : u u w w u σ σ σ σ ρ (4.47) ( ) ( ) w u u u u w u u u u : : : : : : : : : : : : σ σ

84 ha 4. Suaon Mhos 66 Ds PML quaon (4.47) (4.47) u u u u : : : : : ( ) w : : : : : : σ u u u u : ( ) w σ u u u u : : : : : ( ) w : : 3 44 σ : : : : 3 u u u u : ( ) w σ : (4.47f) w [ ( ( ) )] α : (4.47g) w [ ( ( ) )] α : (4.47h) w [ ( ( ) α )] 3 : (4.47) w [ ( ( ) α )] 4 : (4.47) w [ ( ( ) )] α w w w w : : : 3 : 4 w : u u : : u σ σ u u : : : 3 : : σ σ u u : u : : : : 3 u u u : : : u : : (4.47) w [ ( ( ) )] α w : u : u : u : u : : (4.47) w [ ( ( ) α )] 3 : (4.47) w [ ( ( ) α )] 4 w w : 3 : 4 σ σ : : 3 σ σ : : 3 A hnss s hosn fo h PML an h s quaons a n aoss hs gon. h h h PML h o gaua h ang whh sus n ss fon fo whn h PML gon. Nong ha h a fv ua quaons n h no bu 3 n h

85 ha 4. Suaon Mhos 67 PML h a-off s bwn ov absoon an fas suaon uon an h aa ha auss hs a-off s PML hnss. Sou onons In an FDD suaon a sou a b us n onunon wh na onons o nu ng no h ss. h sou a a on an sha bu s ua oos of on sous on h FDD g. h a wo usfu onfguaons: Ha an ansan. Aso of no s ha an nu wavfo us no hav a D offs n D suaons [7]. Ha sou onfguaon In a ha sou h aua san vaus fo h sou oaons a ovwn a ah s wh h s san vaus. hs s s o n n sofwa an an b us o sua an a-wo snaos. h ao awba o hs onfguaon s ha wavs nn on h sou h fo fons o anoh sou oaon w f off of. hs an b av n D b shung off h sou af h nu us s on bng ans. ha s af h fn-uaon nu us h f vaus a s no ovwn. If h sou s a fa awa nough fo an fos hn b h a wav s f ba o h sou oaon has bn shu off an s non-fv. hs s no an oon fo D n sous baus whn s shu off w bo unsab an bgn o gow whou boun. o avo hs ff h sou wavfo us b zo-a fo h an of h suaon af h sou s shu off. h ha sou onfguaon s aua usfu n D baus whn shu off bhavs n a ansan ann. In D a ha sou onfguaon s a goo o fo an uasoun ansu sn fons fo h ansu fa ofn ou n a. ansan sou onfguaon Fqun on wshs o sua an nn us ong fo h fa-f o aoa a an wav wh suanous nzng h suaon sz fo ov ouaona s. o ahv hs a ansan sou an b us [8][9]. A ansan sou us fuf wo a: I us s h f vaus o h s nu wavfo fo h uaon of h nu an us aow nn wavs o ass hough unaff. hs s aosh b onvovng h s nu wavfo wh h us sons of h sou fo a gvn suaon onfguaon. A suaon s fs un wh a s us s a o h sou oaons an h sons of h ss o ha us s o. N whn h aua suaon s un h nu wavfo s onvov wh h o us sons

86 ha 4. Suaon Mhos 68 o ahv ansan wh s ouung h o wavfo. Fu naon as fo aous FDD suaons a gvn n [8] Sab a Whn usng h FDD ho a s us b hosn. hoosng a sa vau nass h nua aua of h suaon whas hoosng a ag vau ass h uon of h suaon. Howv f h s s hosn oo ag h f vaus w gow onna wh ah s an h sus w b non-hsa. h nua sab of h FDD suaon s h ags ossb aas ha w no su n non-hsa onna gowh. fo a gvn s of suaon h a nuous hos fo nng h sab of a s ss. h os oon a h von Nuann ho [][] anng h nua sson aon [97] an usng na -nvaan (LI) ss ho [] [4]. A von Nuann anass s a ou b assung ha h s on on Fou saa oonn sn usua na b ~. hs subsuon faas h aaon of a s Fou ansfo (DF) o h govnng quaons nang h saa vaabs an ung h quaons o a s of uson aons. hn a z-ansfo an b a o h s oan an ~ h oos of z an b oban. h ss s sab whn π π whh a h ossb wavvos whn a s ss. z fo an [ ] h nua sson aon an aso b us o n h ss sab. h aonsh bwn ~ an ~ h angua an saa fquns oung n h FDD suaon s a h nua sson aon. hs s an afa of h FDD ho an s no snav of h hsa sson whos vaabs a an. h nua sson aon an b us o nfo h onon ha a a wavvos ~ us on sua a fquns ~ an v vsa sn a o angua o saa fqun wou na h onna gowh assoa wh an unsab FDD sh. [97] Lna ss ho an aso b us o an ss sab. h FDD sh an b sn as a u-nu u-ouu (MIMO) ss. h ss an b s u as foows whh s n aoan wh aona LI ss ho []: [ ] A[ ] : M: : M: [ L L ] In hs onfguaon s h sa vaab onssng of a san vaus fo h un ( ) an as ( ) ss A s h sa anson a an h o gnvaus of A us on o whn h un fo h ss o b a as agna

87 ha 4. Suaon Mhos 69 sab. h sa anson a w b of h fo A A λ I I. hs as fnng h gnvaus aua onvnn an hos suh as h shgon ho an b us as was shown n [3]. Oh aoahs a b us un an onons suh as N h o ha a( ) A λ a qua. A λ n onunon wh h fa ha a h agona ns of o nvsga h sab of hs aua FDD naon a von Nuann aoah was us sn s h os sagh-fowa. As a s a h hos w us o n h sab fo a D onasng u wh an a h a o h sa onuson ha s oun bow. h ou fs-o quaons us b obn o na h san fs an son-o san quaons. h a shown whou offs g noaon fo s sn os no aff h sab anass. Paa f h sung s a funons of sa hn h aoa ou s o n h qu a v oaon an o a h nu of hs vaus as h suaon s. W fs n h sab s n D fo a h van ass an hn n h fo h os bas D aousa as. hs s w vf b suaon. o h bs of ou nowg nng h n h D as has no bn vous sb n h au. A D hoognous u wh h govnng quaon fo s fo b subsung h fs-o san quaons (4.9) an (4.3) no h fs-o san-nn quaon fo (4.) sung n a ρ son-o quaon fo aon. In h govnng quaons w s an an oas h quaon fo o D whh osons o a D hoognous u wh no a wavvo. h onnuous an s quaons a ρ : : : : : : an [ ] ρ (4.48). (4.49) h sab of h s ss an b n n hs s as b assung ha h souon s an gno wh wavvo ~ hn b ang a z-ansfo n an a DF n sa of h ss. hs assuon s vaa b qung ha h ss b sab fo a

88 ha 4. Suaon Mhos 7 ~ [ π π ] ang h z-ansfo of (4.49) wh { } :z : osonng o a h ossb wavvos n a s ss []. z an {} :z :z :z z ρ :z :z -:z [ ] (4.5) s h z-ansfo so ha h agun nas h oan. N assung ha h souon s a s an wav a saa DF of h ss gvs z :z :z :z z ρ anng h san an sfng ~ [ ] ~ ~ ~ :z :z :z [ ]. (4.5) z z β wh β. (4.5) ρ [ ] ~ z z βp wh P os( ). (4.53) h oos of hs quaon a sab whn z whh ous whn 4 β P. hus ~ Sn [ π π ] 4 β n P 4 β P. (4.54) β a P ~ os P 4 whh gvs hfo ( ) [ ] an [ ] h fs nqua ans ha β an hfo 4 4β 4 β P. (4.55) βp P ρ ang h osv oo an nong ha h has vo s sab a [97]. (4.56) ρ gvs h ρ. (4.57) h son nqua whh s ha β s of so ns wh s o aaas whh an hav ngav ffv os. Sn an a boh osv hfo an ρ us hav h sa sgn. Insng hs onuson s n agn wh [66] whh sas ha wavs w no oaga whn a u f h sgns of hs aas

89 ha 4. Suaon Mhos 7 a ffn. In onuson h FDD sab quns fo a D hoognous u a ha ρ an ha an ρ us hav h sa sgn. A D onasng u wh W bgn as bfo b sng an b oasng h son-o quaon fo o ρ D. Howv w f h son ha an so ha h aa aas a va n sa ha s h a onasng. h onnuous an s quaons a ρ (4.58) : : : : : : : an ( [ ] [ ]) ρ Ang a s- z-ansfo a saa DF an sfng as bfo gvs z :z :z z z wh ρ z :z z z ρ ρ ~ ~ :z :z :z :z ( [ ] [ ]) ~ ~ ( [ ] [ ] ) ~ ([ ][ os( ) ] [ ] sn ( ) ~ ρ. (4.59) (4.6) (4.6) (4.6) z z βp (4.63) ~ ~ β an P [ os( ) ] sn ( ). W hav nou a vson an uaon b wo n β an P sv o an onssn wh h vous sus fo a hoognous u. W an now a { } a β P βp sn w nou h gno as a o onna nsa of as a osn. ha s h agna a of P was a o faa h auaon an os no sn a ng. h oos of h quaon a sab whn z whh ous whn 4 P a β. Sn P [ ] ρ a 4 h vous s a an hfo (4.64) ( ) ρ. (4.65)

90 ha 4. Suaon Mhos 7 hs sab s n agn wh [3] whh was v n a ffn ann. h son nqua a a P P β gvs h onon ha β as n h hoognous as. A D onasng u wh Whn usng o bouna onons n an FDD suaon o nvsga h sson os of an nfn honon sa a s wavvo s a o h no gon as w as o h bounas. H w wsh o n h sab n a D onasng u whn. h onnuous an s quaons a ρ (4.66) an : : : : : : : ρ. (4.67) Ang h z-ansfo an h DF as bfo so ha P z z β gvs ρ β (4.68) an ( ) ( ) ( ) ( ) ( ) ( ) ( ) P ~ sn 4 ~ os 4 ~ sn ~ os 4. (4.69) Nong ha ( ) 4 a P h sab s ( ) 4 4 ρ. (4.7) A D hoognous u wh hs snao sns h ss hsa van D as. h son-o san quaons a fs n b subsung (4.9) hough (4.3) no (4.) an (4.). In hs govnng quaons w s ρ an. h sung onnuous an s quaons a

91 ha 4. Suaon Mhos 73 ( ) ρ (4.7) ( ) ρ (4.7) ( ) ( ) [ ] ( ) ( ) [ ] ( ) [ ] ( ) [ ] : : : : 44 : : : : 44 : : : : : : : : : : : ρ (4.73) an ( ) ( ) [ ] ( ) ( ) [ ] ( ) [ ] ( ) [ ] : : : : 44 : : : : 44 : : : : : : : : : : : ρ. (4.74) h sa ou of ang a oa z-ansfo foow b a saa D DF s s aab ha ah f has s own wavvo. ha s ˆ ~ ˆ ~ ~ s h nua wavvo of h f an ˆ ~ ˆ ~ ~ s h nua wavvo of h f. Bgnnng wh h san f ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) ( ) ( ) [ ] :z :z :z :z :z :z :z :z :z z z ~ ~ ~ 44 ~ ~ 44 ~ ~ ~ ~ ~ ρ (4.75) ( ) ( ) ( ) ( ) ( )( ) :z :z :z z z ~ ~ ~ os ~ os ρ ρ (4.76) ( ) [ ] :z :z N z M z (4.77) wh ( ) ( ) ( ) ( ) ~ os ~ os 44 M ρ (4.78) an ( ) ( )( ) ~ ~ 44 N ρ. (4.79)

92 ha 4. Suaon Mhos 74 A sa anass of h san f s :z :z [ z ( M ) z ] N ~ 44 ~ wh M ( os( ) ) ( os( ) ) an N ρ ρ ( ) ~ ~ 44 ( )( ) (4.8) (4.8). (4.8) hs ou has wo quaons (4.77) an (4.8) n wo unnowns. Fo ~ ~ ~ onvnn on an hoos sn h sab s ah as boh aoah h sa bouns. In hs as N N N. If h suaon aas a hosn so ha nus (4.8) gvs :z :z :z :z [ z ( M ) z ][ ] N[ ] Fo onvnn w an s M an N as foows: M β P wh 44 M M M ρ hn M M M an (4.77) (4.83) z z M N (4.84) β an os( ) ~ P (4.85) M 44 ~ N β N P N wh β N an P N. (4.86) ρ oason wh (4.53) shows ha h oos of (4.84) a sab whn 4 { N} hfo 4 β M n PM a β N a PN a 4 { M N} β M a PM a β N n PN a ~ ~ π π os P Sn [ ] PN N β P N hfo ( ) [ ] M P [ ] an [ 4] M a N a (4.87). h n h son nqua an b gno sn s aoahng zo n h squa sns whas h oh s na aoahng zo. hfo h s an b sa as 4 { M N} Sfaon of h fs nqua gvs 4 4β M 4β N β M PM PM (4.88)

93 ha 4. Suaon Mhos 75 β β (4.89) M N ρ ρ (4.9) 44 ρ ( ) 44 (4.9) ρ (4.9) h son nqua β P gvs h onon ha β whh s ha P 44 has h sa sgn as ρ. M M M 4. h anssson a ho h anssson a ho aus a h anssson an fon nfoaon gang a sng of a D honon sa no on a a h anssson a. h anssson a fo an s s s oos b ung a h anssson as fo h nvua honon sa s. h os of h anssson a an b us o nvsga h sson aonsh an h anssson an fon sa of a honon sa. I s qu aav fo anass baus ovs os-fo souons o hs obs howv s o on nsona anass. h MM aoah has bn a n D oagn sss [5] nabng suh ffs as bfngn an has ahng o b hoough nvsga [6]. Aousa MM ho has bn sb n [7] an has nab na aa o b vf [8]. Nonna ffs an ong w sb n [9]. h ho has aso foun us n o o suus us an nozzs [] an n snusoa ouga ubs []. h anssson a ho fo a honon sa aus h sang an oagaon nfoaon n saa as an hn obns h o fo h anssson a. Bgnnng wh sang Fgu 4.3 shows h nn an sa san wavs a h nfas bwn ans of ffng aas n a D honon sa. In h as of a ouga ub wavgu h sa s an an sonnu whh ous a h bounas bwn ffn oss-sona aas as was sb n Son.3.3. h fon offn fo an nn san wav avng fo aa o aa s ( ) ( ) ( ) ( ) an fo aa o aa s. Fo h san wavs un onsaon so h anssson offns a gvn b ( ) an ( ) M.

94 ha 4. Suaon Mhos 76 a b a b Fgu 4.3. Nonau us o sb san wavs a h bounas of a D sa. A sang a as h nn an sa wavs as a b S S S S a b. (4.93) Sang as a onvnn baus h s of h a sb anssson an fon. A h nfas bwn ffng gons h sang as a gvn b S S. (4.94) Anav a anssson a as h wavs on on s of a sa o h wavs on h oh: b b whh an aso wn n a noaon as a a (4.95) b a. h ffs of u sas an b han ogh s b ung h osonng anssson as. h sang an anssson a foas an b onv usng h foowng aons whh a v fo h a fnons: Nong ha S fo boh S an n as S S S. (4.96) S S S whh an b wn n n s of as h anssson as fo ah nfa an b (4.97). (4.98) Wavs an b oaga saa b a san hough a u b usng h foowng oagaon anssson a:

95 ha 4. Suaon Mhos 77 ( ) P. (4.99) hs sson assus ha h oagaon u s ossss an ha a wav funon s gvn b ( ) wh. h anssson a nasuas a h sang nfoaon whn on an s n b squna obnng h ffs of sang an oagaon as ( ηa) ( ( η) a) P P. (4.) hus h anssson hough on n of honon sa fo a aua assung a on h f ns son an a on o h gh s gvn as a a. Fo n s onn ogh h anssson a s gvn qu s as ha s unoua n h sa as (.. ) an hfo usng h ho sb n []. n. I shou b no n an b qu foun o a h anssson offn h anssson a us b onv ba no a sang a usng (4.96). h S vau osons wh h anssson offn fo on n of h o h oh an s gvn b S a. (4.) ηa n n h anssson hough n s s S wh s h () n of h a. n 4.. Ea auaon h vous sus an b a o h ouga ub sb n boh [7] an n Son.3.3 o n h anssson su hough h ub as h nub of sgns s nas. h anssson su na h fs ban ga s shown n Fgu 4.4 fo a ub wh a 85 η an 6. As h 343 s ban ga anuaon nass as o sgns a aah. h as shown n Son.3 w aso aua usng h MM.

96 ha 4. Suaon Mhos 78 anssson offn - - Sgns Fqun (Hz) Fgu 4.4. aua anssson su fo a D ub wavgu showng ha as o sgns a a o h ub h ban ga anuaon nass [34]. 4.3 Mu sang ho h anssson a ho sovs h ob of u sang b saang h aous wav sang an oagaon whn a honon sa. Mu sang ho uss a sa aoah s aab n hgh nsons. Muh of hs son s n fo h sus suaz n [] whh s an n sou on h o. Aong sf fns o [] w no b a sn h fo h bass of hs sus. An aoah an o u sang ho was us b Lo agh n hs fs a on h sub n 89 [3] an was a foaz n 93 b ávša [4]. I was fuh vo b Faan [5] an Waan [6] [9]. Mon aous an oagn MS has ogss fo ha-sh os o sa-fng onas [3] ha aow onnuous aa aas. Aous MS [3] an s aaon o o suus [3][33] hav bn nsv sb n h au. I has bn n fo a saaf ho o aoun fo h vo nau of as wavs [34] an has bn a o as shs [35] an as [36]. Mu sang ho has bn a o nuous unqu aous onfguaons nung aas of baoons [37] ano [38] an o a [33] [39]. Iovng h ffn of MS agohs b unang h s of bass funons has aso bn sb [4]. Subwavngh agng has vn bn nvsga usng MS as a ans of sbng hs hnonon [4][4].

97 ha 4. Suaon Mhos Sang fo on n In wo nsons ons a hoognous fu u wh ns an bu ouus ρ an K sv. a ha κ K wh s h soo as sffnss offn (Vog noaon) an κ s h aaba ossb. Fuh s ossss an ssonss has a onguna has vo K an a haas aous ρ an ρk. Eb whn hs hoognous u s a fu n of aus whos n s oa a h ogn. Fo h bng w a sng ou anass o fus so ha ansvs wavs a no ons an a saa f a fu sf h wavs n quson. h oonas a h aa ( θ ) o asan ( ). h f na o h n an b v no an nn f n ( ) an a sa f s ( ). H w hoos o b h saa vo ona. hus h oa na f s h fs ob h wav quaon ( ) ( ) ( ) n s. (4.) ρ K. (4.3) onsng on -haon oon wh u( ) fqun an u( ) { } s h angua u s a o-vau saa nvo h ob s u o fnng souons of h Hhoz quaon ( ) u (4.4) wh s h wavnub. In na oonas h Hhoz quaon has saa souons ψ θ θ ( ) H ( ) an ψ ( ) J ( ) ˆ Ζ (4.5) a ougong an gua na wavfunons sv. h ougong wavfunon ψ ( ) sasfs h Sof aaon onon a nfn bu s sngua a h ogn whas h gua wavfunon ψˆ ( ) os no sasf h Sof aaon onon an s no sngua a h ogn. h nn f fo a an wav a an ang α o h oona as s u n ( α sn α ) os ( θ α ) os. (4.6)

98 ha 4. Suaon Mhos 8 wh αˆ an αˆ s a un vo n h on of ang α. No ha w us h fnon of a an wav. Ang h Jaob anson w osϕ J ( ) w ϕ ( ) as (4.7) w an w h nn f as an anson of gua wavfunons a a uo anson: u n ψˆ h nn f offns ( ) wh α an [ M M ]. (4.8) a nown an n hs as oson o a an wav. h vau of M shou b hosn o b ag nough o nz h ouaona o wh sa nough o ou h suaon n a asonab aoun of. W hoos o sn h sa f as an anson onssng of ougong wavfunons o sasf h Sof aaon onon: u s ( ) h sa f offns a h qu unnowns. ψ. (4.9) So fa w hav sasf h Hhoz quaon b oosng h fs of s souons w hav sasf h Sof aaon onon b hoosng o us ougong na wavfunons o oos h sa f an w hav a nown gua na wavfunon ooson of h nn an wav. Now w wsh o sov fo h sa f offns as a funon of h nn f offns so ha o n n. (4.) In (4.) { } an { } f offns an { } n n a oun vos onanng h sa an nn s a h anson a o s us -a. No h sub nonau ffn bwn h anson a us h an h anssson a us n a on nsona anass. h -a w b foun b ang bouna onons a h nfa bwn h sa an h suounng u. A soun-ha n A a soun-ha nfa h Nuann bouna onons a so ha u on S (4.) n wh S nos on h bouna of h sa. Fo a n of aus hs bos

99 ha 4. Suaon Mhos 8 u Sn u u u ψˆ ( ) ψ ( ) n s. (4.) hfo ψ (4.3) ( ˆ ( ) ψ ( ) ) θ θ ( J ( ) H ( ) ) θ ( J ( ) H ( ) ) (4.4). (4.5) Baus of h ohogona of h onna faos h suaon saas no M nnn quaons: ( ) H ( ) J. (4.6) hfo fo soun-ha na sas h ns of h -a a wh n n J H δ s h Kon a. No ha F ( z) ( ) δ n ( ) of h Bss o Han funons (s 9..3 n [43]). (4.7) ( z) Fn n n Fn ( z) Fn ( z) wh F n s an z z h Kon a n (4.7) s nowoh baus ans ha h vaous os of h na wavfunons a o ou. hs s baus w a onsng ns whos fs naua oos no na wavfunons so h -a s agona. If h sa was non-na h -a wou no b agona an h saa os wou b ou. h aa aon of hs s ha -a a s b sn n sofwa as a vo of offns. A soun-sof n A a soun-sof nfa h Dh bouna onons a so ha whh ous a u on S (4.8) fo a n. Foowng a sa vaon as bfo as o J ( a) δ n ( a) n. (4.9) H

100 ha 4. Suaon Mhos 8 A nab n In h vous wo ass h n was nab. Fo a nab n on us sov h osonng anssson ob. h n s ns bu ouus has vo an an wavnub a ρ K an sv. h f whn h n an b sn as a gua na wavfunon sn s no sngua a h ogn an os no n o sasf h Sof aaon onon as h f on ns o h bouna of h n. h f v us aso obsv h Hhoz quaon whn h n ( ) v an hus an b wn as ( ) b v ψˆ. (4.) wo anssson bouna onons us b a h nfa bwn h n an h suounng u: v u on S (4.) an n v n u γ on S. (4.) wh γ s so offn nng on h fnon of u an v. Fo ou anass (4.) bos v u ρ ρ. (4.3) W hav wo quaons (4.) an (4.3) wh wo unnowns ( b an ). Fo (4.): ( ) ( ) ( ) ( ) ˆ ˆ b ψ ψ ψ. (4.4) Fo (4.3): ( ) ( ) [ ] ( ) ˆ ˆ b ψ ρ ψ ψ ρ (4.5) ( ) ( ) [ ] ( ) J b H J θ θ θ ρ ρ (4.6) ( ) ( ) [ ] ( ) J b H J θ θ θ ρ ρ. (4.7) B ohogona of h onnas ah us qua zo. aangng fo b : ( ) ( ) ( ) J b H J ρ ρ (4.8)

101 ha 4. Suaon Mhos 83 b [ J ( ) H ( ) ] ρ ρ J ( ). (4.9) No ha ρ ρ ρ ρ ρ ρ h ohogona of h onnas s wh. Subsung (4.9) no (4.4) an agan nong ( ) H ( ) [ J ( ) H ( ) ] J β (4.3) Fna aangng gvs h -a as n J H 4.3. Sang fo wo ns ( ) ( ) J β. (4.3) J ( ) βj ( ) δn ( ) β H ( ) ons wo soun-ha ns wh h h aus gvn as h oaon b s h ogn. h oa f s wh. (4.3) ( ) ψ ( ) ψ ( ) an ns oa a b. u un us ψ ˆ (4.33) b an h suss no h sa nub. o sov hs ss of quaons s nssa o ss (4.33) abou a sng ogn so ha h onnas us a oon ang an h s of h suaon a ohogona. Fo h nn wav hs s s aosh b anng h nn f abou on b as u n ( ) ψˆ (4.34) Fo h sa fs h ansaon of ogn s a ossb b af s aon ho whh fo ou uoss an b wn as wh b ( ) ( b ) ψ ( ) ψ ˆ (4.35) Sn n n b b o o oa n a noaon as wh ψ { ψ ( )} ψˆ { ψˆ ( )} an { ( )} ogn h a n ψ S ψˆ (4.36) S. Sn b S s a a saaon a an aows a na anson abou an b o b -an abou an ogn b. No ha (4.36) s on va fo < b

102 ha 4. Suaon Mhos 84 whh s awas va n h as of non-ovang ns sn a ns a gvn qu s as Sn ( b ) ( b ) n < b. h saaon ψ. (4.37) Pung a ogh (4.33) a b wn abou boh b an b as ( ) ˆ ( ) ψ ( ) S ( b ) ψ ( ) n n u ˆ n ψ (4.38) ( ) ˆ ( ) S ( b ) ψ ( ) ψ ( ) n n u ˆ n ψ. (4.39) h nn suaon a b aang so ha a h wavfunons a of h sa o: u ( ) ˆ ( ) ψ ( ) ψ ( ) S ( b ) n n ˆ ψ (4.4) n n ( ) ˆ ( ) ψ ( ) S ( b ) ψ ( ) u ˆ n n ψ. (4.4) h bouna quaons (soun-ha o soun-sof) a now b a o (4.4) an (4.4). If h ns a nab h w b an aona s of offns an an aona s of bouna onons as subsqun sb. wo soun-ha ns Ang h soun-ha bouna onon (4.) o (4.4) gvs u ( ) ˆ ( ) ( ) ˆ ψ ψ ψ ( ) Sn( b ) n (4.4) n J θ θ θ ( ) H ( ) J ( ) S ( b ) Sn h onnas a ohogona v of h suaon us b zo J ( ) H ( ) J ( ) S ( ) n n n n n (4.43) b n (4.44) ( ) H ( ) J ( ) S ( ) n J b (4.45) whh s h fs s of quaons. B h sa ou h son s of quaons s ( ) J ( ) S ( ) H ( ) J n b n. (4.46) h abov wo quaons sb ( M ) quaons wh ( ) n n n M unnowns.

103 ha 4. Suaon Mhos 85 wo soun-sof ns Pong as n h soun-ha as bu usng (4.8) as h bouna onon ( ) ( ) ( ) ( ) ( ) ˆ ˆ n n n S u b ψ ψ ψ (4.47) ( ) ( ) ( ) ( ) n n n S J H J b θ θ θ (4.48) ( ) ( ) ( ) ( ) n n n S J H J b (4.49) ( ) ( ) ( ) ( ) n n n S J H J b. (4.5) B h sa ou h son s of quaons s ( ) ( ) ( ) ( ) H S J J n n n b. (4.5) hs aso oson o ( ) M quaons wh ( ) M unnowns. wo nab ns Mofng (4.4) an (4.4) o nu an f nna o h n gvs ( ) ( ) ( ) ( ) ( ) ( ) n n n b S u ˆ ˆ ˆ b ψ ψ ψ ψ (4.5) ( ) ( ) ( ) ( ) ( ) ( ) n n n b S u ˆ ˆ ˆ b ψ ψ ψ ψ. (4.53) Ang h fs bouna onon (4.) o (4.5) ( ) ( ) ( ) ( ) ( ) ( ) ˆ ˆ ˆ n n n b S u b ψ ψ ψ ψ (4.54) ( ) ( ) ( ) ( ) ( ) n n n J b S J H J θ θ θ θ b (4.55) ( ) ( ) ( ) ( ) ( ) n n n J b S J H J b (4.56) ( ) ( ) ( ) ( ) ( ) J b S J H J n n n b. (4.57) h son s of quaons s ( ) ( ) ( ) ( ) ( ) J b H S J J n n n b. (4.58)

104 ha 4. Suaon Mhos 86 Ang h son bouna onon (4.3) o (4.5) gvs u ( ) ˆ ( ) ( ) ˆ ( ) ( ) ˆ ψ ψ ψ Sn b n bψ ( ) (4.59) n J J J h fna s of quaons s θ θ ( ) H ( ) θ ( ) S ( b ) b J ( ) n n n θ ( ) H ( ) J ( ) S ( ) b J ( ) n n (4.6) b n (4.6) ( ) H ( ) J ( ) S ( ) b J ( ) J n b n. (4.6) ( ) J ( ) S ( ) H ( ) b J ( ) n J n b n. (4.63) n hus (4.57) (4.58) (4.6) an (4.63) sb 4 ( M ) quaons wh 4 ( M ) unnowns. A wo-n souon usng -as If h -a fo ah n s nown ( ) a us -a o -sua an b fn ha aus a of h sang so ha o o o. (4.64) W bgn b onsng h f nn on S onssng of h nn f an h sa f fo S : ˆ ( ) S n( b ) n ψ. (4.65) n Sn s nown an haazs h sang of S an nowng ha (4.65) s h f nn on S hfo w an w ( S ) (4.66) whh uss h noaon fn n (4.36) an wh h ansos oaon ass fo h hang n suaon fs nou n (4.4). Sa ( S ). (4.67) Enang fo (4.66) an sfng: ( S ( S ) (4.68)

105 ha 4. Suaon Mhos 87 Sa (4.69) S S S ( I S S ) ( S ) (4.7) B ( S ) wh ( I S S ) B. (4.7) B ( S ) wh ( I S S ) N h oa sa f an ss as B. (4.7) Sˆ Sˆ (4.73) wh h as Ŝ a saaon as va fo > b. h ov-bas no o onugaon. h fnon s sgh ffn han n (4.37): ˆ ( b ) ψˆ ( b ) an S ˆ ( ) S n n { } Sn b ˆ. (4.74) h nn f a h vaous n ns an b ss n s of h nn f n abou h ogn as ˆ S. (4.75) Fna subsung (4.75) no (4.7) an (4.7) hn boh no (4.73) gvs h - sua. ˆ ˆ ˆ ˆ ˆ ˆ o S B S S S SB S S S. (4.76) Sang fo N ns ons N ns wh ns oa a Eah n has a sa f an h oa sa f s b an h h aus gvn as [ N]. u s ψ ( ). (4.77) N hus fo soun-ha o soun-sof ns h s on bouna onon n sung n a ss of N ( M ) bouna onons wh ( M ) N ss of unnowns ( N ). Fo anssson obs h s aso an aous f whn ah n v b ( ) ψˆ (4.78) aong wh wo bouna onons n. hs ans h a ( M ) onons wh ( M ) N ss of unnowns ( N an b b N ). N bouna

106 ha 4. Suaon Mhos 88 W a s gnaz h sus fo h wo-n ass o av a h N-n quaons. nazng (4.45) o gv h soun-ha quaons N n n n J ( ) H ( ) J ( ) S ( b ) [ N ] nazng (4.5) o av a h soun-sof quaons N n n n J ( ) H ( ) J ( ) S ( b ) [ N ] nazng (4.57) an (4.6) o h nab n quaons N ( ) H ( ) J ( ) S ( ) b J ( ) n n n. (4.79) (4.8) J b (4.8) N ( ) H ( ) J ( ) S ( ) b J ( ) J b. (4.8) n n n An N-n souon usng -as h vous sus an b gnaz o an nub of sas whos -as a nown. nazng (4.7) fo N sas N B S hs an b wn n h fo wh N B I S S. (4.83) wh B δ I ( δ ) S. (4.84) N [ ] sng a gnaz (4.75) an (4.73) gvs h -sua as N N ˆ o S S. (4.85) 4.4 h an wav anson ho h an wav anson ho s a w sabsh ho fo nng h sson os of wavs n sos [44]. I has bn w us fo nng h on ban suu of sas an has subsqun bn a o hoon an honon sas. Fo honon sas h san f an h aa aas a oos no an wavs ha a o wh s o h sa a. h Boh onons a hn a o h bounas of a un whh sus n an gnvau ob ha vas h ˆ

107 ha 4. Suaon Mhos 89 sson aon. ansvs wav os a sn n h souon baus h as wav quaon s us. h vaon sn bow a obns h aoahs of [45] an [46] hough shou b no ha [47][48] ov n bagoun. W assu ha h sa s oos of ossss ssonss a. h PWE ho has bn sussfu a o D honon sas [49][5] nung anng wavgung an f os [5] honon sas wh ansoo os [5] sufa wavs on D sas [53][54] h on of gan ban gas n fu-fu honon sas [55] ban gas n fu-so honon sas [56] ban gas n so ns susn n a wh fs [57] as wh o nusons [58] an sufa [59] an bu [6][6] aous wavs n zo honon sas. I has aso bn a o 3D suus nung o as ooss [6][63] ansoo hoon sas [64] fu-fu honon sas [65] g sh-a honon sas [66] an sbng h ng bans n on [67]. I has aso bn of o nu vansn os an us fo fnng h anssson an fon a an nfa [68] o o fo ouaona obs a fu-so bounas [69] an o ov h onvgn whn ang wh sonnus [7] Dvaon of h an wav anson ho n 3D As n ow nsons an as sb n Fgu. a 3D honon sa s an aa wh a un ha s o n h nsons. h un ou hav onnuous os suh as a na gan fo on aa o h oh aa whn h un. h PWE ho n h nsons s as u o D b ovng anous s an faos. W bgn wh h as wav quaon n h nsons usng Laé offns nsa of h sffnss s w hav us vous. hs s o an onssn wh h noaon an vaons n [45] an [46]. Subss an fn h nson ρ s h ns an λ an µ a h Laé offns of h anssson u a of whh a funons of sa. h as wav quaon s u u λ ρ u µ u Sn h aas ρ λ an µ a funons of wh ( z) w anno a h as onsans as w hav n a hoognous aa.. (4.86) s a asan oona N w fn a s of oa a vos (LVs). hs w b us n h onna bass funons of h saa Fou ansfos. h an b nvson as h

108 ha 4. Suaon Mhos 9 saa quvan of h s fquns of h Fou ansfo of a s- sgna. In h quaons a s h h vo of h un an s an ng: an b a a 3 π a ( a a3 ) (4.87) b b b 3b3 a a 3 π an a ( a3 a) b a a 3 π. (4.88) a3 ( a a ) No ha h nonao s qua n v as baus of h nau of h saa ou whh osons o h vou of h un. h aa aas a an as a saa Fou ss usng h LVs n h onna o fn h s of bass funons. ρ ( ) ρ λ ( ) λ an ( ) µ µ (4.89) wh f s h agnu of h h Fou oonn n h anson of h funon f. Aong o Boh s ho [95] h souons o h wav quaon w b oua snusos of h fo ( ) u ( ) wh u ( ) u n a 3 ha s ( ) u (4.9) n. (4.9) u s un- o. hus w an an h funon on h sa un- bass s as bfo ( ) u ( ) u u. (4.9) In hs quaon s s o bng whn h fs Boun zon a on ha s subsqun suss n ga a. Vaus of ha a ous hs gon a b ah nsa b ang a LV o a sa ha s whn h fs Boun zon whh s a onsqun of aasng u o h o of h ss. o an (4.9) fuh Boh s ho sas ha ( ) u s o wh s o h sa a an hfo w an us h sa bass s of onnas as bfo fo h aa aas n (4.89). An nuv anaon of hs su of Boh s ho s as foows. Sn h sa un onsaon s nfn h s no wa o sngush on un fo anoh b av oaon; hfo h wav fqun oonns us b h sa n v un an on h has hangs bwn s. If h sa was fn hn ah ou b sngush b s oson an Boh s ho wou no ho.

109 ha 4. Suaon Mhos 9 Now ha w hav ohogona oosons of h aa aas an wav souons a usng h sa bass s w a now subsu h Fou ansons no h wav quaon. h goa s o g boh ss of h quaon o hav h sa onn ( ). In h foowng noaon ( ) ans h oonn of vo sov n h on ( ) ( ) ˆ. h s n bas su fo ang a ona vav of h o onna. Fo a ( ) ( ) ( ) ( ) [ ] ( ) ( ) z z. (4.93) W bgn b subsung h Fou anson of u no h wav quaon (4.86) u u u u µ µ λ ρ. (4.94) N h anson fo u s subsu no h quaon ( ) ( ) ( ) ( ) u u u u µ µ λ ρ. (4.95) h sbo s ans ha h suaon on h HS s no o h suaon on h LHS. h boh f o h s of LVs. oung h nnos vavs ( ) ( ) ( ) ( ) ( ) ( ) ( ) u u u u µ µ λ ρ. (4.96) Eanng h Laé offns ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) u u u u µ µ λ ρ (4.97)

110 ha 4. Suaon Mhos 9 wh h Laé offns hav bn an abou hosn so ha. obnng h onnas o an ou ( ) ( ) ( ) ( ) ( ) ( ) ( ) u u u u µ µ λ ρ. (4.98) oung h fna vav ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) u u u u µ µ λ ρ. (4.99) Fna h ns offn s an as ρ ρ wh w now hoos o n h HS wh h LHS gvng ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) u u u u µ µ λ ρ. (4.) obnng h onnas o an ou ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) u u u u µ µ λ ρ (4.) whh an b aang as ( ) ( ) ( ) ( ) ( ) ( ) ( ) u u u u µ µ λ ρ. (4.) h abov quaon us b va fo a hfo v n h bas us b zo fo v ho of whh s an gnvau ob of h fo Λ A. In hs onfguaon A s a a of offns s a a of gnvos an Λ s a agona

111 ha 4. Suaon Mhos 93 a of gnvaus. Bgnnng wh h A a fo a aua an ( an ) h ns of h a a oos of 3 3 as so ha ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) I A 3 z z z z z z z z z z z z µ µ λ ρ (4.3) wh 3 I s h 3 3 n a. onsng a una s of M LVs h oa a A s oos of h sub-as an s [ ] M L A A. (4.4) h a of h gnvos s z M M M z z M M M z M M M M M M u u u u u u u u u u u u L M L M L (4.5) an h a of h gnvaus s 3 M O Λ (4.6) wh 3M 3M Λ A. h M 3 gnvaus oson o h ffn ng bans of h honon sa. h gnvaus sn h squa angua fquns aow fo ah hosn an hfo aow on o n h sson os of h honon sa. h fs Boun zon In 3D h fs Boun zon s boun b ang h sufa nosng a sng a a n (a h Wgn-Sz an fn b a a an a 3 ) an snng ha

112 ha 4. Suaon Mhos 94 sufa n oa sa (fn b h LVs). N s an on n h oa a as h ogn an onn o nghboung ons. onsu h nua bsos of hs ns. h gon of nson s h fs Boun zon. [7] h onsuon of h fs ub Boun zon s shown n Fgu 4.5 fo a D angua a an n [46] fo a fa-n ub n 3D. h ons na on h fgu sn ons of ns. h aow fquns aong a onnuous ah of -vos ha onns so o a of hs ons s o whh ans h sbos ofn foun on h boo as of sson agas. In D h ah s fo Γ ( ) o X ( π a ). In D h ah s fo Γ ( ( ) ) hough X ( ( π )) an M ( ( π a π ) o Γ. a a ) an ba Fgu 4.5. h onsuon of h fs ub Boun zon fo a angua a n D nung h ao a ons of ns. h v angua a vos a shown n a) an h v vs a vos a shown n b). In ) h hosn ogn s onn o s nas nghbous. h nua bsos of hs ns s shown n ) as fnng h bouna of h fs ub Boun zon an n ) an ab wh h ao ons of ns. Eng bans wh as a a aous bans an h ohs a a oa bans. Whn h a an ough of h ow an hgh ng bans fnng a ban ga agn wh ah oh h ban ga s a. If h o no agn s a n Inaon as Dung suaon h a wo ossb aoahs: h fs s o ou h Fou offns a ah s whh sfs nng bu sus n nuous unan auaons. h son ho s o -ou a h nssa offns an us so nng ho o a whh offns a n a ah s. Eh o han M offns w b nssa o vaus ous of a an ang w b una. If M s an o nub ub hn h qu nub of -ou offns w b ( ) 3 3 M. hs vau s av a b obsvng ha wo LVs a suba whn oung h bass

113 ha 4. Suaon Mhos 95 funon of h Fou offns. Sn h s of M LVs a onan n a ub n nvs sa v obnaon of subang wo of h w su n a ub of suba LVs wh a s ngh oub h ogna. On hs vaus a ou an so h on n o b n whn oung h ns of A whh s an naon-sf as. In gna h s of LVs a fo a hob s nsa of a ub bu h n ans h sa. Fo D sss h qu nub of -ou Fou offns w b M. h 3D quaons a b u o D b ng v h ow an oun of a A whh ssna ovs h z nson fo h wav quaon. If -oung of h offns s qu an M s an o nub squa hn h qu nub of ou offns w b ( M ). 4.5 Oh suaon hos h a oh suaon hos ha w no o n h ous of hs hss wo an so of hs hav bn sussfu a o h anass of honon sas. h nu a u sang suosa hnqus an h fn-n ho (FEM) n aon o oh hb o unqu os an ouaona hos. h a sb bf h fo h sa of onss. h a u sang ho [7] [7][3][33][7] [79] s a hb bwn h ons of u sang ho an h anssson a ho. Mu sang ho s us o n h -a fo a aua un of sang aa suh as a sab of o sas. N h -as fo u sabs a obn hough hos v sa o h anssson a ho aowng ag-sa suus o b oos ou of sa-sa sang suus. Sn h a hos sn n Son 4.3 aow h onsuon of suh aoso sang suus on a agu ha w hav aa sn a u sang. Howv h hos sn n h fns a qu foaz an ohnsv an gv sf nsuon fo ang wh o sas (s. s Equaon (7) an (8) of [33]). Psuosa hos [8] [84] us a saa Fou ansfo o sa h saa vavs n h wav quaon aowng h Fas Fou ansfo (FF) agoh o b us. Saa vavs aua as n h FDD ho a sub o nua sson an a o has o a hgh saa fquns. hus hgh souon gs a qu o ga h nua sson a hgh fquns sung n ong suaon un s. Psuosa hos ss hs ob b auang h vavs n h -sa oan (h saa-fqun oan) wh has o s no sn. Howv h awba

114 ha 4. Suaon Mhos 96 s ha non-o suus a ffu o sua baus of h nhn o of h FF. h fn-n ho s sa o h FDD ho ha nga fos of h unng PDEs a us nsa of ffna fos as n h FDD ho [8]. h suaon gon s aon no ns an h f vaus a h bounas of hs ns a us o f a oa bass funon o ah n. Bouna onons a a o h g of h fn-n suaon gon as n h FDD ho. hn h qu nua ngaon s fo aoss h ns o n h f vaus n quson. h fn-n aoah has bn a o suh vs obs as nvsgang sufa sonan sas an sunsng n honon sas [33] a u hoon-sa ahz ass [83] an aaas [84] [86]. Wh s o honon sas h fn-n ho has bn us o su sufa aous wavs n bu honon sas [87] an honon sa as [5] [7][88][89]. Fna s woh nonng ha h suaon an ong hos sn a nh hausv o usv. S [9] fo a vw of sva of h hos sn n aon o so ohs no suss h. So hos obn faus of sva ffn hos suh as n [9] wh h auhos us a fn ffn of h govnng ffna quaons fo h gnvau ob. In [9] an FDD aoah was us n -sa nsa of asan sa. A u-ass o was us n [] an [93] o v h ban suu of honon sas. A ansf-a aoah usng n's funons was o n [94]. In [95] a has-sa o was us. A ho a gno ahng ho whh bas a song sban o h PWE ho was sn an us n [96]. Fna wav-bas hos w us n a an-wav anson n [97] as an anav o Fou-bas ansons. I shou b a fo hs as ha h a of suaon an ong hos fo o aous sss s qu h. 4.6 Anab o -vang aa aas o nvsga whh of h suaon hos sn a anab o anazng vang aa aas on us fs ons h funana aons of suh hangs. onsng h wav quaon wh a -vang ns n D an bgnnng wh onu F u P (4.7) P u u. (4.8)

115 ha 4. Suaon Mhos 97 onvng hs o h ns fouaon an subsung h onsuv quaon gvs u ρ u σ ρ (4.9) u σ (4.) u ρ u u ρ (4.) whh s h of D wav quaon fo a -vang ns. hs fouaon wou aa o suo h noon ha a -vang ns ofs h wav quaon bu fuh onsaon as o a ffn onuson. On ou agn a ub of aa avng a a gvn vo wh no na fos. If h ass of hs ub sohow nass h nw aona ass wou n o sha h n ng of h na ass an h vo wou as. Howv hs agun os no s o ho fo a vang ns. If wo ubs of qua ass an ns a avng a a gvn vo squzng h wo asss ogh no h sa ou b on of h na ubs wou oub h ns. Howv sn ah ub ha h sa na n ng h s no ng ansf whn h wo ubs a squz ogh. So wou s un hs usans ha a -vang ns an a -vang ass a no quvan ons h fo a onu sv o ng onsvaon sv. ρ o u hs o h ns s hangng bu ass s onsv so. hs s ossb baus ρ V an hfo h hang n ns ognas fo h hang n vou an no a hang n ass ρ V V ρ V V bu (4.) V. (4.3) hs shows h su ha a ngav hang n vou as a osv hang n ns whou affng h ass of h ss. hus h wav quaon s no funana a b a -vang ns. h wav quaon ans unhang whn onsng a -vang ossb ( ) as w. hs s baus h a no -vavs sn n h sss-san

116 ha 4. Suaon Mhos 98 u aonsh an hfo σ ( ) ( ). Sa aguns aso a o h sss-san aonsh n hgh-nsons. In gh of hs aguns h s nohng funana ohbv abou an of h sn suaon hos n h sn of -vang aa aas. h quson s h how oa s h naon an wha bnf wou suh a o an suao ov. h FDD ho an b us as-s s b uang h aa aas a -s n h suaon. Howv s suffs fo h sa bnfs an awbas as bfo. Na anno ov os-fo souons fo h bhavou of a -vang ss an hfo os no gv an nsgh no wh a aua bhavou s oung. h MM an MS hos an b of o han -vang aa aas an w gv os-fo souons o wav oagaon n -vang suus. hs s h bu of nw sah sn n hs hss n has 5 an 6. h PWE ho ou b of o han -vang aa aas bu hs avnu has no bn nvsga baus h o of h gnvau ob obfusas h anaa aonshs of h ss.

117 ha 5 DYNAMI PHONONI YSALS IN ONE DIMENSION h honon sa ffs vous sb a a naowban an hgh nn on h sz an sang of h sas whn h sa os ha a f an unhangab. W hav oos ha a b ossb o naa a h bhavou of honon sas b vang h aa aas n ossb b usng av aas. In hs ha w n h sng D sa honon sa ho h anssson a ho a n Son 4. so as o han -vang aa aas an aso o ov nsgh no h faos govnng -vang honon sa bhavou. sng h anssson a ho as a sang on h anssson quaons fo boh onnuous (nung non-o) an o -vang aa aas a v. W hn vo a os-fo souon o h aous wav anssson hough a D vang honon sa ha shows n agn wh FDD suaons an wh uon s sva os of agnu fas. W hav a ou nw D ho h vang anssson a ho (V-MM). sus show ha anssson os an b sgnfan a usng -vang aa aas as h vng hans. Effs suh as aa afaon an sgna swhng a onsa n suaon. Fna so onusons a awn. 5. h -vang anssson a ho o vo h quaons fo a -vang honon sa s nssa o fs ons on h anssson hough a sng -vang. A honon sa an b a vang b ang an of s fnng aas va n. Howv hs anass w fs ons on a -vang an. In h na as a -vang ub a n a ouga ub wavgu s a suoga fo vang aa aas n a honon sa. h ub wavgu was hs ha onans aa ubsh n h 9 a b DW Wgh an S obbo aang n Sa Maas an Suus 8 58 n Aous wav anssson n -vang honon sas

118 ha 5. Dna Phonon sas n On Dnson sfa hosn as a o baus has h o ha s onsan vn whn h ub a hangs so ha on h ffv aous an hangs. hs s quvan o h ns an bu ouus of a aua aa n a honon sa havng a onsan ao sn ( ) K ( ) ρ ( ) bu h an s f o hang sn ( ) ρ ( ) K ( ). h su s ha h oagaon as a -nvaan ( P an P ) bu h anssson as ( ) an ( ) a -vang. hs son was na hosn o a h ob aab. Subsqun w hn gnaz ou ho o nu -vang oagaon as. 5.. anssson hough on fng o Son 4. w n h vaabs an as o b -vang. L [ ] [ A A ] ( ) a ( ) a ( ) a (5.) ( ) ( ) ( ) an { a ( )} A( ) wh I nos h Fou ansfo. Aso A (5.) I (5.3) { ( )} ( ) I (5.4) so ha h agun nas whh h funon s n h o angua fqun oan. -vang sang h ns of a -vang whn s a funon of. Assung ha s fqun nnn h -vang sang anaogous o h sng fqun as shown n Fgu 4.3 s gvn as ( ) ( ) a( ). (5.5) b hus a onohoa wav nn on a -vang sa w b oua sung n a sa su as n Fgu 5..

119 ha 5. Dna Phonon sas n On Dnson Fgu 5.. A onoon s sa an oua a v -vang nfa. -vang oagaon Whn h ans oon of a onohoa wav asss hough aa whos has vo s -vang h wav w ungo has ouaon as shown n Fgu 5.. Fgu 5.. A onoon s oua as oagas hough a -vang sab of aa. wo ouang osss ou whn a aa ha has a -vang has vo. h fs s ha h nsananous wav nub ( ) a h nfa whn aa bos a funon of h nsananous has vo ( ) an h nu angua fqun suh ha ( ) ( ). (5.6) h son ouang oss s ha h oagaon fo a wav o av a san hough aa bos a funon of ( ). I an b vsuaz as a onvo b of ngh wh an nsananous s ( ) h onvo b a ( ) ang aous a sans ha n an h onvo b a. In hs anaog h aa sf s no bng anso on h sans fo on n o h oh. hs son oss s fa o. A oason of h vaous auaon hos avaab s gvn n [98] an hs nu anaa s-anaa an nua aoahs. Fo h bng w ons on onsan has vos an vo a subsqun susson o hs ob.

120 ha 5. Dna Phonon sas n On Dnson P h -vang oagaon a s ss n h angua fqun oan ( ) an an b unsoo as a a of how uh ng of an nu fqun s oua o an ouu fqun. hus ( ) ( ) A( ) B P. (5.7) h -vang anssson a In h foowng von a ( ) an ( ) a a saa b a san a an b an a s na oaons as sb subsqun. o vo h -vang anssson a w o fo ( ) a on h f ns of son (s Fgu.5) o h gh as n h vous sa as. Assung a onsan has vo oagaon fo h son nan o h fs nfa s gvn as B ( ) ( ) A ( ) P (5.8) wh h san oaga s ( η)a. Poagaon fo son no son obs h anssson an fon fn b h anssson a howv hs a s now a funon of. Sn h nfa s nfnsa sa h s no oagaon a an h wavs on h gh-han s of h nfa an b wn as ( ) ( ) b( ). (5.9) Poagang h wavs fo hs nfa o h n an b wn as wh h san oaga s son s D ( ) ( ) ( ) (5.) P ηa. Fna oagaon fo son ba no ( ) ( ) ( ) a. (5.) obnng a of hs quaons s h wav oagaon hough on gvn b n whh A { { { } ( ) I ( ) I ( ) I ( ) I { ( ) ( ) } P P A (5.) I nos h nvs Fou ansfo. Eanng h Fou ngas obnng h no onnas an ong h ngaon gvs A 4π 4π 4π ( ) ( ) ( ) ( ) ( ) A ( ) P ( ) ( ) ( ( ) ) ( ) ( ) ( ) ( ) A ( ) ( ) ( ) ( ) ( ) A ( ) P P P P P (5.3)

121 ha 5. Dna Phonon sas n On Dnson 3 wh an f o h nu an ouu angua fquns sv. hs sfs o ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) 4 4 π π A A A P P (5.4) wh ( ) ( ) ( ) ( ) ( ) P P. (5.5) 5.. -vang has vo Wh a -vang has vo ( ) h oagaon as w b of h fo n (5.7) whh s h ang fo an nu fqun o an ouu fqun. onsqun h abov quaons an b wn usng aona vaabs of ngaons as foows. Equaon (5.) bos ( ) ( ) ( ) ( ) ( ) ( ) { } { } { } { } F F F F A A P P (5.6) wh h subss n h Fou an nvs Fou ansfos na whh vaab s nvov n h ansfo. Eanng h Fou ngas obnng h no onnas an ong h ngaon gvs ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) π π π A A A A P P P P P P L L (5.7) wh h ss on h fs n nas ha h quaon s onnu bow fo sa asons. hfo ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) π π 4 4 A A A P P (5.8) wh ( ) ( ) ( ) ( ) ( ) ( ) ( ) P P. (5.9)

122 ha 5. Dna Phonon sas n On Dnson 4 auaon of P As fo h aua auaon of ( ) sab of aa of hnss wav nng h gon s θ ( ) φ wav a ha oaon s fn as P on a ons h has of a wav ng a wh a vang has vo ( ). If h has of a n wh φ aouns fo an has offs hn h n θ. h has ng h gon θ ( ) ( ) ( ( ) ) φ ou w b θ (5.) ou wh ( ) s h oagaon fo a wav o av fo on n o h oh. I s sf a funon of baus h has vo s hangng. hus s h souon o h quaon ( ) ( ) τ. (5.) τ h ansfoaon fo nu o ouu s hfo an so h su a h ouu s ( θ θ ) ( ) (5.) ou n ( ) { } I. h oagaon a s hfo h ouu su u b h fqunnn has shf noun b ah fqun oonn an s ( ) ( ) I { } P. (5.3) o h ha h has ng h gon wh a onsan has vo s h h nga of (5.) s fo wh onsan gvng su θ ( ) φ ou Subsung hs no h quaon fo h has a h ouu gvs whh s h su of a has a of an as o h oagaon a θ ou. (5.4) φ (5.5) φ an s quvan o h a [66] ( ) πδ ( ) P. (5.6) If h has vo s vang snusoa hn w a w ( ) [ vos( ) ] (5.7)

123 ha 5. Dna Phonon sas n On Dnson 5 wh s h -avag has vo v s h vaaon fao an angua fqun. In hs as (5.) s wh [ v os( τ )] s h ung τ (5.8) ( ) v τ sn ( τ ) (5.9) τ ( ) v ( ) [ sn ( ) ( [ ( ) ])] sn (5.3). nfouna hs quaon s no as sovab fo ( ) aoaons o sfaons a nssa. A h v as ( ) nua usng h quaons ov n hs son. P an so an b n 5..3 anssson hough u s W w now v ba o h assuon of a onsan has vo. o n h aous wav oagaon hough u -vang s w bgn b onsng h anssson hough wo s an hn w n hs su o n s. Fo an nu su of A ( ) h ouu su fo h son an b oban fo (5.4) as A 4π ( ) ( ) A ( ) n ( π ) ( ) ( ) A ( ) wh n fo hs as. ong h ngaon as o A [ ] ( A ) n ( ) ( π ) ( ) ( ) n ( π ) ( ) A ( ) wh ( ) ( ) ( ) (5.3) (5.3). Enng hs su o ss h aonsh bwn h nu an ouu sa a h wo ns of an n- -vang honon sa s wh ( ) n n ( ) ( π ) ( ) A ( ) A n n (5.33) s h -vang anssson a fo n s an s gvn b n ( ) ( ) ( ) n n n n (5.34)

124 ha 5. Dna Phonon sas n On Dnson 6 n whh n s an na ngaon vaab. Fgu 5.3 usas oagaon hough n -vang s. Bgnnng wh whh has h os-fo souon gvn n (5.5) an b onsu b a ngaon usng (5.34). If ah s na hn n n a b onsu onna ( 4 8.). Howv f h s a no na o a ungong ffn aa aa vaaons hn aaon of (5.34) usng h osonng o ah unqu. n a b onsu b a Fgu 5.3. h -vang anssson a ovs an ouu su fo a gvn nu fqun onvson ba no a anssson su o a h anssson offn s nssa o onv h anssson a ba no a sang a. h onvson foua gvn n (4.96) s va fo h vang as f on onss h nsananous oan souon fo a aua nu fqun ov whh o sov h sang a. Sng gvs ( ) ( ) ( ) n n S n ( ) (5.35) n ( ) n wh S ( ) I[ ( )] an ( ) I [ ( )] A ( ) n S n n. Fo an nu sgna su of n h anssson ouu su fo h n h an b n b ngang ov a nu fquns.. hus h funon ( ) n A ( ) S ( ) A ( ) n. (5.36) n S an b hough of as a sufa wh ah on as how uh of an nu fqun s onv o h ouu fqun whn n -vang s a onn ogh. 5. Souons whn h aa vaaon s o W a aua ns n suaons whn h aa aas va oa n whh sns a o of oaon ha has no bn aqua o. H w assu ha h has vo s onsan o h sf h anass. Howv usng h sus of Son 5.. o souons fo -vang has vos ou aso b nooa.

125 ha 5. Dna Phonon sas n On Dnson Sng- souon Agan assung a onsan has vo us n h sson fo ( ) gvn a nown o aa vaaon sgna (.. h ns of an a nown o funons of ). Sn an a o h an b sn b h Fou ss ( ) n an ( ) n n (5.37) wh s h funana aa aa vaaon angua fqun (oon a h ung fqun) an Fou offns. h Fou ansfos a hn gvn as n n n a h osonng as of o n n ( ) δ ( ) an ( ) δ ( ) n n n. (5.38) ang (5.5) an subsung h nown Fou ss sa ( ) ( ) ( ) ( ) ( δ P n δ n P ). (5.39) Ingaon ov h of h a funons n (5.39) sus n ( ) ( ) ( ) ( P n n P δ ( n) ) n n. (5.4) hus gvn an nu fqun an a ung fqun h van ouu fquns ov an b n b hoosng a an n o a h agun of h a funon n (5.4) qua o zo osonng o h onon: ( n). ha s hoosng an n ns whh ouu fqun vs ha aua nna onbuon fo h nga whh s now ss as a suaon. o faa a -fqun fouaon w hang h ns o a so ha n q n (5.4) q n q. hs aows o s h fqun of an q o faa h suaon of a van as of an n osonng o ha fqun. sng hs subsuon an nong ha nabs h anssson a fo a sng o b ss as ( ) ( ) q ( q ) ( ) δ ( ). q P q P (5.4)

126 ha 5. Dna Phonon sas n On Dnson An n- souon h abov sng souon an b n o n-s b fs obanng h anssson a fo wo s. h wo a s whn h squa bas of (5.3) a a wh suaons as gvn b (5.4).. ( ) ( ) ( ) ( ) an. Nong ha an so ha ( ) h ngaon n (5.3) an b a b ( ) ( ) ( ) ( ). (5.43) Foowng a sa von o ha us n Son 5..3 fo an n- wavgu h anssson a fo an nu fqun of an a ung fqun of s gvn b n ( ) ( n ) ( ) ( ). (5.44) 5..3 h s anssson su Fo a onohoa nu h ouu su w b a s s of fquns onssng of h nu funana an su an ffn fquns wh s o h ung fqun whh w hoos o a haons fo h sa of bv. Wh h h of (5.36) an ang ha h s ouu su an b wn as A ( ) S ( ) A ( ) n. (5.45) n onsqun h ouu su onans a ss of a funons a h van fquns. ang (5.35) w s a os-fo souon fo ( ) S fo h nown o Fou ss offns of ( ). ouaon of h ( ) n n S n -oan funon s a ahvab fo suaon uoss bu s no sab fo anass. In auang ( ) S n h qu vson b h n Fou ss obfusas h aonsh bwn h sang an anssson Fou offns. Fnng h Fou offns of ( ) s saghfowa an h Fou offns of ( ) n n a s gvn n ( ). Howv fnng h Fou offns of ( ) ( ) n fo h onsun Fou offns whou oung h vson n h oan s ah o hangng. In hs ga h ho sb b Duffn [99] fo auang h Fou offns of h oa of a o o funon s of n n

127 ha 5. Dna Phonon sas n On Dnson 9 oan. h fs s n h anass s o a fo (5.35) h () n of h sang a ng Is Fou ansfo s gvn b wh ( ) ( ) S S ( ) ( ) ( ) n n n. (5.46) { } ( ) I ( ) D( ) n n (5.47) D n whos Fou offns n o b foun. h Fou ansfo n (5.47) s saghfowa sn h agun s sf a Fou ss whh an b an n s of h vaous Fou offns. A a ho fo an nvs Fou ss ho 4 n [99] nabs h Fou offns of D ( ) no b h vo Α { } o b foun fo h Fou offns of ( ) no b h vo F α { f }. h ho an b sa as s a a n whh g wh { } souon fo h Fou offns of ( ) n ( ) n n Α F (5.48) an g f f an b foun fo o aa vaaon sgnas.. hus a os-fo S n s of h nown Fou offns of h su ss b (5.48) qus ha h nvs of a a b n whh an bo non-va as h nub of haons un onsaon an onsqun h sz of nass. Howv fo h uos of vong a os-fo quaon o anaz h ffs of aa vaaon on aous wav anssson on a hoos o an on h nfun of h funana an fs haon. In hs as s a 3 3 a so ha s nvs a b foun. Fuho whn onsng h anssson of a sgnas on bos a osv fn Han oz a whh fuh sfs h os-fo ssons. A aoah fo a s offs snuso h as of Fou offns n (5.37) an b aua fo h nown Fou offns of h -vang ans f h a s offs snusos. h Fou ss fo a o funon ( ) s ( ) n n n. (5.49)

128 ha 5. Dna Phonon sas n On Dnson ons a s offs snuso fn b wh ( ) ( φ) os n gna sn h an s a an su of h oa s ( ) D an has nfn an oonns gvn b D ( ) n n n D n α n (5.5) φ. If D( ) ( ) h n (5.5) n φ D (5.5) so v offn s nown f D an α an b foun. W an aua α fo h fs wo offns D D α (5.53) hfo D α n. (5.54) D an h ob us o fnng D an D. L a an b. Fo n [3] h D offn s Fo 5..3 n [3] D. (5.55) a b a D. (5.56) b a b hfo α n a b b D n D a a b. (5.57) hus h as of Fou offns n (5.37) an b sf n n s of h an Fou offns. 5.3 os-fo souons W wsh o fn h anssson offn S n fo h quaons fo so ha w a gan nsgh no h faos govnng aous wav anssson hough -vang

129 ha 5. Dna Phonon sas n On Dnson honon sas n D. W ons on h funana an fs haon o sf h anass an show ha hs s a goo aoaon fo sa aa aa vaaons. W ons h as wh s onsan bu s vang oa n wh an angua fqun of. ha s ( ). (5.58) W hav assu a onsan has vo A os-fo sson fo on W wsh o fn h funana fqun anssson su fo on of a vang honon sa. o o hs w us sov fo an hus h sang an oagaon as of ns nung h funana an fs haon. Sn ( ) ( ) w us fn souons fo ( ) a h vaus [ ]. h -vang sang as Fo (4.98) an subsung n h fnon of n s of (5.59) an. (5.6) Sn (5.6) hfo h as of Fou offns a nown as an (5.6) wh n gna. o fn n w a us offns fo h nvs of an offs snuso gvn n (5.55) an (5.56). Fo bv w sha av h offns as n D. hus D D D D S S (5.63) wh ( ) ( ) D. hfo h as of Fou offns a

130 ha 5. Dna Phonon sas n On Dnson D D D D D an D. (5.64) h -vang oagaon as H w usa a usfu o of h oagaon as. I aows a fs-haon oagaon a o b ss n s of h oagaon as of h funana an ung fquns. Sn ( ) ( ) ( ) P (5.65) hfo ( ) ( ) ( ) ( ) ( ) P P P P P. (5.66) h a Subsung no (5.4) an sang s onanng faos oh han h funana o fs haon gvs ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) P P P P P P P P P P P q q q q (5.67) ( ) a a b b D D D D D b b 4 (5.68) wh ( ) a a a θ η a b η an a η. Fuh sfaon vnua as o

131 ha 5. Dna Phonon sas n On Dnson 3 ( ) ( ) a a D D D b b b b sn sn os os θ θ θ θ. (5.69) h a Foowng a sa von as bfo w subsu no (5.4) an sa s onanng faos oh han h funana o fs haon gvng ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) P P P P P P P P P P P P P P P P q q q q (5.7) ( ) a a b b D D D D D b b b b D 4 (5.7) vnua sfng o ( ) ( ) a a D D D D b b b b b sn os os os os θ θ θ θ θ. (5.7) h a H w subsu no (5.4) gvng ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) P P P P P P P P P P P q q q q (5.73)

132 ha 5. Dna Phonon sas n On Dnson 4 ( ) a a D D D D b b b b D 4 (5.74) ( ) ( ) a a D D D b b b b sn sn os os θ θ θ θ. (5.75) h anssson offn fo on -vang o fn h funana fqun anssson offn ( ) S w us subsu h vous aua as no (5.46) wh ( ) ( ) ( ) ( ) ( ) M S (5.76) wh ( ) ( ) M an ( ) (5.77) wh ( ) as was vous aua fo [ ]. N anng h nan of h a an sfng ( ) ( )( ) ( )( ) (5.78) ( ) ( ) ( ) ( ) ( ) ( ).(5.79) onsng on h funana an fs haon of ( ) M h a b aua usng h nvs Fou ss ho n (5.48) bu no h ho n (5.55) an (5.56) sn ( ) a b o an s no a s offs snuso. h funana an fs haon of ( ) an b nvson as an s on h o an aaz b

133 ha 5. Dna Phonon sas n On Dnson 5. h ho of (5.48) o a 3 3 a s fa aua ov ha h s s suffn sa fo h ogn o o u anoh wa ( ) s an offs snuso whos vau s suffn sa zo. Fo o aa aa vaaons h a fas Fou ansfo (FF) us b fo on an ov-sa vson of h -vang aa aas o ov aua o h ho of (5.48) a b us nung h hgh-o zo offns. In an as fo anaa uoss w s us M o ua h o of ( ) (5.76) an sfng gvs ( ) M ( ) ( ) M M M (5.8) n h anssson offn. Subsung hs vau no ( ) ( ) ( ) ( ) ( M M M ) W on wsh o now h D oonn of S ( ) an ang o S (5.8) so w hoos s whos onnas ( ) ( ) M ( ) ( ) M M (5.8) whh s h os-fo quaon fo aous anssson hough on of a -vang honon sa A os-fo souon fo wo an n s o fn h os-fo sson fo h funana fqun anssson su fo wo s of a -vang honon sa w us n whh vaus of a qu n h sson fo. Sn h os-fo souons fo funana an fshaon vaus of w gvn n (5.69) (5.7) an (5.75) h anson of s f n s of. W agan ons on h funana an fs haon of. Fo (5.43)

134 ha 5. Dna Phonon sas n On Dnson 6 ( ) ( ) ( ) ( ) ( ) ( ) s s s s s s (5.83) wh s an s a subsuons o faa a -fqun fouaon. onsng ha ( ) (5.84) w hfo qu an wh ( ). h a foun b nsng h s aguns no h suaon n (5.83) an sang s wh faos oh han h funana an fs haon. sng h abbva noaon ( ) ( ) h h qu as a ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) s s s s s s (5.85) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) s s s s s s (5.86) an ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) s s s s s s. (5.87)

135 ha 5. Dna Phonon sas n On Dnson 7 h anssson su an b foun b usng h sa quaon n (5.8) usng h ns of h as an wh M ( ) ( ). In fa ( ) foun b usng (5.85) (5.87) o ov h anson fo ( ) a s a b n an anssson su wh h o subsuons as bfo. n S an b n n (5.44). h as. Equaon (5.8) a b us agan o fn h 5.4 Mhooog an vfaon A a wahough of h an ss of h nw V-MM fouaon s gvn h. I s hn shown ha h n ho shows a ogah gowh n ouaona o as h nub of -vang s s nas as oos o a na gowh n oaab FDD suaons. Fna sus fo h n ho a oa agans sa FDD suaons an show n agn ouaon ho ( ) ( ) ( ) ( ) n ( ) ( ) n ( ) S n ( ) ( ) n S n Fgu 5.4. A fowha usang h ss an quaons nssa o sua wav oagaon hough a -vang ub wavgu. Dash bos na h vaabs an as an so bos na h quaons an oaons qu. a ans ha (5.44) shou b a as an s as nssa o ahv n -vang s. A ou fo nua auang h anssson su s usa n Fgu 5.4 o h gu h a hough h an ss nvov. Bgnnng wh h -vang ans ( ) an ( ) as ( ) an ( ) a subsu no (4.98) o fn h -vang anssson. N h as of o Fou offns n an a foun s b ang an FF o h anssson as o b usng h ho fo h nvs of an offs snuso gvn n (5.55) an (5.56). h sng- -vang anssson a ( ) an hn b foun fo (5.4). If h a u -vang s hn (5.44) a b a as an s as s nssa o gna

136 ha 5. Dna Phonon sas n On Dnson 8 n ( ) fqun whh aouns fo anssson hough n s. hoosng a aua nu an fong an nvs FF s ( ) n. Fna onvng ba no a -oan sang a usng (5.35) an hoosng h () n gvs h oan anssson offn ( ) S n aua nu fqun na ( ) S. n. A fna FF gvs h anssson su fo a I shou b no ha fo h uos of suaon (5.48) was no us sn h uh s FF an nvs FF a b us n s a. Howv fo sa vaus of aa aa vaaon s aua nough n s 3 ( ) 3 fo fo fnng ( ) S fo n. Aso no ha h son of hoosng a aua nu fqun n s f whn vng h gna souon sn h Fou ansfo an s nvs o no n o b fo h o of auaon hough s ossb o squna aua 3. u o n ffn o a an onna aoah: 4 h -vang anssson as onna nung a { ( )} s fa o. o o hs w bgn auang [ ] og n wh. (5.88) In h abov quaon L ans h foo oaon whh ouns own o h nas ng. Anoh wa o sa hs s o aua ( ) b ows of wo u o h au aho n h bna snaon of n. Fo a f n ( ) 4 an 8. n hn aua h na as a hn obn so ha h a u o h o n. Baus h as hav bn aua as ows of wo h qu as ha n o b obn va ngaon oson o h oaons n h bna snaon of n wh h gs a s. sng h abov a of n w wou obn ( ) an ( ) h s ( ). o av a hs fna s on qu on ngaon sn h w wo s n h bna snaon of n. In gna h nub of ngaons qu w b h nub of s n h bna snaon of n (a h Hang wgh of n an no as n ) nus on. hus h oa nub of ngaons qu o gna ( ) gnaon of ( )) no b ( n) s gvn b n 8 (no nung h

137 ha 5. Dna Phonon sas n On Dnson 9 ( n) n n og. (5.89) A o of ( n) vsus n shown n Fgu 5.5 a shows a ogah n as. hs s aua nowoh gvn ha h sz of a oaab -vang FDD suaon wou gow na as o s a a. hus hs n ho s suo o h FDD ho fo suaons nvovng ag nubs of -vang s. Fgu 5.5. A o of h nub of ngaons qu fo n -vang s. I an b sn ha h o foows a ogah n as h nub of vang s s nas. A oaab FDD suaon un- wou gow na wh h nub of -vang s Vfaon As no a h FDD ho s ab o han -vang aa aas n o suus. As suh was us as a bnha n o o vf ha h nw V- MM fouaon s o an o oa suaon s. A sng -vang was anaz usng h FDD ho an h V-MM fouaon usng h ho usa n Fgu 5.4. h ouga ub was sgn o h sa sfaons as n h a auaons n Son 4.. an n [7]. Aona h oss-sona aa ( S ) of son was va snusoa b ± 5% a a fqun of 6 Hz an h nu aous wav was a. Hz onoon. In o o vn fons fo a ha sou n h FDD suaon a ansan sou naon was us [8]. Shown n Fgu 5.6a) s ha h sung anssson sa fo on -vang a n n agn. h so aows na ha h ash ns a a funons. Sn h nu onoon s a a funon has fquns a ±. Hz. h as a Hz oson o h son haon of h ngav nu fqun ( Hz 6 Hz Hz ). Aona haons a sn bu hav agnus ha a oo sa o aa n h fgu. h onnuous nau of h FDD o s u o h unaon of a fn sgna

138 ha 5. Dna Phonon sas n On Dnson wh an fqun oonns. In gna s ffu o n a onsua sang o ha wou sooh una a h haons sn n h sa sgna. Anoh onbung fao s ha h suaon bgns wh a aous a sans qua o zo so us b aow fo sonans o bu u n o o aoa h sa-sa as au b h V-MM fouaon. In s of hs onsans s a ha h anssson as sn n h FDD suaon hav as ha oson a o h V-MM suaon. Fuho h V-MM suaon uon s w fou o fv os of agnu fas han h FDD suaons us o gna Fgu 5.6. Sho FDD un s a ossb b ung h sua uaon an b ung h sa an souons. Howv hs uons su n a oo fqun-oan souon ga susb o sgna ossng afas an ss aua u o nas nua sson. Shown n Fgu 5.6b) a h anssson sa fo n na -vang ub s whos aas a h sa as fo Fgu 5.6a). H agan h V-MM fouaon s sn o b n goo agn wh h FDD suaon. In h -vang as h sn of anssson offns a shows ha a sng nu fqun ous a u of fquns a h ouu. hus a na obnaon of nu fquns auss a na obnaon of ouu sa whh s nasua n (5.36). Fgu 5.6. oason of anssson sa usng h nw ho an a FDD ho fo a ung fqun of 6 Hz an an nu fqun of. Hz. a) Sng -vang. b) n -vang s. Aows na h sn of a funons. Baus h FDD suaon was fo ov a fn o h su s onnuous. Ha h suaon bn fo ov an nfn ong o h su wou hav bn a funons a h a oaons gvng n agn fo h wo ss of sus.

139 ha 5. Dna Phonon sas n On Dnson 5.5 sus h funana fqun anssson offn ( n (5.44)) fo h n ub s s shown fo boh h sa an -vang ass n Fgu 5.7 gna b vang an ong h sung ouu su. h sa o n Fgu 5.7 s h sa as ha shown n Fgu 4.4 fo a n- sa ub. In h na o h aas a h sa as n Fgu 5.6b). h anssson su now nus ng ha has bn oua o an fo h haons b h -vang anssson an fon offns. No ha h shas of h ban ga an ban ga gs hav bn sgnfan a bong ss oun an o gua han n h sa as. hs ass h ossb of hangng h ban ga haass usng -vang aa aas as h onong hans. On ou nvson a onoab noh f usng suh an aangn. anssson offn Fgu 5.7. Funana fqun anssson offns fo n sa an -vang s. In h na as h -vang aa aas oua h nn wavs a v an bouna ausng ng o b onnuous oua o an fo h funana fqun. h ung S s vang b ± 5%. fqun s 6 Hz an ( ) h sa o s ou n Fgu 5.8 wh a ung fqun of 4 Hz. h hang n ung fqun has a hang h anssson su fo ha shown n Fgu 5.7. Whas h ban ga anuaon was sgh u n h 6 Hz as s nas n h 4 Hz as.

140 ha 5. Dna Phonon sas n On Dnson anssson offn Fgu 5.8. Anoh funana fqun anssson su fo n sa an -vang s. hs s h sa as n Fgu 5.7 bu wh a ung fqun of 4 Hz. No h aa ffn a b h hang n ung fqun. Wha s no shown n hs wo fgus a h haons sn u o h ouaon oung a ah nfa. Fgu 5.9 shows h funana an fs haon anssson sa hough n na -vang s. h onons w na o h na as n Fgu 5.8 an h bu n osons o h funana fqun su of boh fgus. S f ou (Hz).. f n (Hz) Fgu 5.9. anssson sa hough n -vang D honon sa s nung h funana an fs haon. h bu n osons o h na su n Fgu 5.8 an o n (5.44). h gn n s an h n s. h a nfn an suh ns n aon o h h shown Paa afaon W no ha an ung fquns su n onna gowh of h FDD suaon f vaus an ha ovng h an sa souon an nasng h. 3.

141 ha 5. Dna Phonon sas n On Dnson 3 suaon uaon no ga h ob. Iovng h souon s o naow h ang of ung fquns ha wou aus onna gowh. I was hfo sus ha h gowh was a funon of h ss an no a funon of h FDD suao. vn h oa -vang nau of h ss an ff a aa afaon was sus a h fquns of onna gowh [6]. I s w nown ha an ung fqun of n wh n s a osv ng w su n aa afaon bu as n s nas h aoun of ng onf o h ss ass. As suss n Son.3. sng -vang sss ho on as wh o n h h sa o nsons bu no n boh suanous. onsqun h sang on of ou anass was b ans of h FDD suao whh qu no funana hangs o han -vang aa aas. On ob wh usng h FDD suao o nvsga -vang honon sas s s nab o ou a u onohoa nu n a fn-uaon suaon; hn h av soohnss of h os n Fgu 5.6. hs s aso wh was ffu o soa h aua fquns sus of oung aa afaon sn h banwh of h ung an nn fquns a funons of h suaon uaon. S (B) f (Hz) Fgu 5.. A sufa oos of h funana fqun anssson sa fo vaous ung fquns. A fon vw s shown n a) an h o vw s shown n b). h s osonng o f s h sa anssson su fo h ouga ub wavgu n [7]. h an of h sufa osons o h anssson su fo S ( ) vang b ± 5% as h ung fqun s nas. No h o of h bhavou n boh h f an f ons.

142 ha 5. Dna Phonon sas n On Dnson 4 In onas ou n ho ovs souons fo a fquns whh s a o aua an nsghfu anass of aa afaon whn -vang honon sas. Fgu 5. s a o of h funana fqun anssson su as a funon of ung fqun. h s of h o aong h f ou an osons o h sa anssson su n Fgu 5.7 an Fgu 5.8 an h ss aong fou 4 Hz f ou 6 Hz an oson o h na os n Fgu 5.7 an Fgu 5.8 sv. h ons ha gow an anua sha a ons of nfn gowh an anuaon whh an b sn whn fuh suaons a un o zoo no hs aas of h o. I was foun nua ha h aas of h o osonng o nfn gowh ou whn s zo whh as o M n (5.8) (h avag of ( )) an ( ) onons nssa fo aa afaon S bng nfn. h fnon of M s o fuh o n un wha onons gos o nfn. Sn w a onsng on h funana an fs haon M π π π π ( ) φ φ φ φ n (5.9) I I I wh ( ) ( a a )( osφ sn φ) ( a a ) ( a a )( osφ sn φ) (5.9) an a a I. W an w hs as π M φ π I (5.9) wh ( a a ) os ( a I ai ) sn φ a an I ( a I ai ) ( a a ) φ ai φ (5.93) os φ sn. (5.94) M w go o nfn whn h nonao of (5.9) gos o zo fo so ang φ whh ous whn boh ( φ ) an ( φ ) onvnn fo an I a suanous zo. Essng an I n h o A os φ B sn φ (5.95) I I I I A os φ B sn φ (5.96)

143 ha 5. Dna Phonon sas n On Dnson 5 wh h offns an b n s b oason wh (5.93) an (5.94). o sov fo h ang φ a whh boh s of h nonao a zo w w (5.95) an (5.96) n a fo as whos souon s Sn sn ( aos( ) ) hn h s a φ [ π ] A A osφ sn φ A B an f Bosφ (5.97) Bsn φ A B B A sn os B A os sn. (5.98) (5.99) ha sasfs (5.98). I s a fo nua suaons usng h nw -vang ho ha hs s u a ons of nfn gowh. Sn hs ons of nfn gowh a oung n boh h FDD suaons an n ou nw -vang ho a h sa ung fquns an o no ou n h sa as w onu ha vang aa aas an aus aa afaon un h gh onons Sgna swhng On ossb aaon of h of anssson os n -vang honon sas s swhng a W sgna b hangng h ung fqun. o onsa hs on wo ung fquns w hosn ha ha oos ffs on h anuaon n h n of h fs ban ga of h -vang ouga ub wavgu. Fgu 5. shows h sa fo f 8 Hz an fo f 4 Hz. h anssson whn h ban ga s hang b aoa B aong o h nw -vang ho.

144 ha 5. Dna Phonon sas n On Dnson 6 anssson offn ~B Fgu 5.. h anssson sa fo wo ffn ung fquns hosn o onsa swhng of a Hz W sgna. h s aoa a B ffn bwn h anssson offns a Hz. N an FDD suaon was un wh an nn Hz W wavfo an a ung fqun ha ana bwn 8 an 4 Hz. Fgu 5. shows a sho- Fou ansfo of h ans sgna ha onsa h swhng ab wh a ow na ang. hs s u n a o h fa ha h fn nau of h ung sgnas ans ha h w no f naowban as n h nw V-MM fouaon. hus h ung sgna su was sooh an no s n h FDD suaon. Sa h nn us was aso no a s fqun bu was u wnow o u sgna afas. A hs suaon non-as an ha ng s sn a oaons of h sufa n Fgu 5. oh han h wo ons sb b ( f ) ( Hz8 Hz) ( Hz 4 Hz) f an sung n a ow swhng na ang han n h a as usa n Fgu 5.. Nvhss hs onsaon of W sgna swhng as b ou n ho nas ha -vang honon sas a b ab o swh fo a ansssv o fv sa s b hangng h ung fqun of an av aa.

145 ha 5. Dna Phonon sas n On Dnson 7 Hz anssson (B) Fgu 5.. h noaz sho- Fou ansfo of a Hz ans sgna fo an FDD suaon of a -vang honon sa swhng bwn h wo sas usa n Fgu 5.. h ns na whn h ung sgna was swh. h na ang of aoa 4 B s ss han h B u o non-as n h FDD suaon. h nn wavfo fqun s n h n of h fs ban ga. 5.6 Sua an onung susson W hav sn a nw fouaon fo ng h anssson os of -vang honon sas. hs ho gvs os-fo souons o h anssson hough sng an u -vang s. In aua a o aa-vaaon sgna sus n a s suaon of haons. A v sang nfa h -vang aa aas oua h ng of h oagang wavs o an fo h funana fqun sung n aaa a anssson os. I was shown ha h n ho os ahs h ons of FDD suaons wh uon s an os of agnu fas. I was aso onsa ha -vang aa aas an aff h nau of aous wav anssson n honon sas. hs ffs an o b onsa na an h ho an sus sn n hs ha nas ha suh fuu sah a b wohwh an fufu.

146 ha 6 DYNAMI PHONONI YSALS IN WO DIMENSIONS In hs ha w us u sang ho o n h -vang sus o na sas n wo nsons. Ou anass s bas on h saa vo ona hb ng h ho o fu-on hoss an sas. W oa hs -vang u sang ho (V-MS) o FDD suaons fo boh on an u na sas. I s shown ha h wo hos ou sa sus bu ha ou n ho s suo n an was. Mos noab h ouaona o nass ogaha as oos o na wh h FDD ho ang fas whn a ag nub of sas a nvov. Fna so onusons a awn an ossb ovns a suss. 6. -vang u sang ho Ou anass uss h sa noaon an assuons as h sa as sn n Son 4.3. Fs w ons a sng -vang n wh onnuous -vang aa aas. h souon s sz an an n -a s oban ha nus h ffs of oung bwn nn an sa oa fquns. Fna h sa f of u -vang ns s nvsga an s shown ha h a quaons fo u sas ssna an unhang. 6.. Sang fo on -vang n I s assu ha h hos aa aas an a onsan. Fuh s assu n h von of h -vang -a ha h sa has a onsan has vo bu a -vang an ( ). hus ρ ( ) ( ) an ( ) ( ) K. h ason fo hs son s ha soas h ffs of -vang aa aas o h bouna onons a h hos-sa nfa. Wh hs son s sn n ou von of h -vang -a fo an nvua n s no sn whn obnng u -vang -as o n h oa ss sang. Oh hos a b us o fn h -vang -a fo h o oa as of a vang has vo. hn h sung -vang -as a b obn usng h

147 ha 6. Dna Phonon sas n wo Dnsons 9 hos subsqun sb. Vaous hos fo auang h sons of ns wh -vang has vos a vw n [98]. h -nn nn an sa fs an b wn n h h o fqun oan. Fo a h nn f s ( ) ( ) ( ) n u ψ ˆ (6.) wh u s h vo ona an h oub aow nas h Fou ua. ha s h oan f an b sn as a su of a su of ohogona na wavfunons. hus h oan funon an b wn as h nvs Fou ansfo of h su as ( ) ( ) ( ) ψ π u n ˆ (6.) nong ha h suaon an nga a b hang u o h ohogona of h s. h bouna onons us b sasf a a s an hus an b sa n h sn of -vang aa aas whn h n as ( ) ( ) ( ) [ ] s n v u u (6.3) an ( ) ( ) [ ] ( ) ( ) s n v u u ρ ρ (6.4) wh v s h vo ona whn h n. Subsung h sa no h fs bouna onon (6.3) an sfng: ( ) ( ) ( ) ( ) ( ) ( ) ˆ ˆ b ψ ψ ψ π π π (6.5) ( ) ( ) ( ) J b H J π θ. (6.6) Baus h onnas θ a ohogona (6.6) bos nnn quaons ( ) ( ) ( ) J b H J. (6.7) Agan baus h onnas a ohogona h nn sson us b zo fo a so ha h nga s zo fo a ang o

148 ha 6. Dna Phonon sas n wo Dnsons 3 ( ) ( ) ( ) J b H J (6.8) whh s h fs bouna onon n h -vang as. h sfaon of h son bouna onon w o as bfo bgnnng b subsung h sa no h son bouna onon as gvn b (6.4). h oan uaon of h as n h bas n (6.4) an b wn as a fqun oan onvouon ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) I I ψ ψ ψ ρ π π π b P b P b P b P b ˆ ˆ ˆ 4. (6.9) wh ( ) ( ) P ρ. If h ns s an offs snusoa funon of h ho sn n Son 5..3 a b us o fn h Fou ss of s oa. Pong wh h subsuons ( ) ( ) ( ) ( ) ( ) ( ) ( ) ˆ ˆ 4 b P ψ ψ ψ ρ π π π (6.) ( ) ( ) ( ) ( ) J b P H J ρ π π θ. (6.) Baus h onnas θ a ohogona (6.) bos nnn quaons ( ) ( ) ( ) ( ) J b P H J ρ π. (6.) obnng h ngas

149 ha 6. Dna Phonon sas n wo Dnsons 3 ( ) ( ) ( ) ( ) J b P H J ρ ρ π. (6.3) ang h vav whh s nnn of h ngaon ( ) ( ) ( ) ( ) ρ ρ π J b P H J. (6.4) Agan baus h onnas a ohogona h nn sson us b zo fo a gvng ( ) ( ) ( ) ( ) ρ π J b P H J. (6.5) Nong ha ρ hs an b wn n s of h haas aous an as ( ) ( ) ( ) ( ) π J b P H J (6.6) whh s h son bouna onon sf. Nong ha (6.8) an b wn as ( ) ( ) ( ) H J J b (6.7) w an subsu (6.7) no (6.6) o av a ( ) ( ) ( ) ( ) ( ) H J B H J (6.8) wh ( ) ( ) J J P B π. (6.9)

150 ha 6. Dna Phonon sas n wo Dnsons 3 hs fao an b wn n a fo sa o (4.3) ha ( ) B s a o h nvs of β baus of h o n whh w obn h bouna onons. aangng gvs ( ) ( ) ( ) ( ) ( ) ( ) J B J H B H (6.) whh a b wn as ( ) ( ) ( ) ( ) ( ) ( ) δ δ J B J H B H. (6.) hs s h onnuous quaon ang h nn an sa fs n h -vang as. Vfaon of h sa as o vf ha (6.) oass no (4.3) un sa onons w s hoos ( ) ρ ρ an hfo ( ) { } ( ) ρ πδ ρ I P. Fuho w ons on a sng nn fqun so ha ( ) ( ) δ π an ( ) ( ) δ π. Subsung hs vaus no (6.) gvs ( ) ( ) ( ) ( ) ( ) ( ) δ δ δ δ J B J H B H (6.) ( ) ( ) J B J H B H (6.3) ( ) ( ) ( ) [ ] ( ) ( ) ( ) [ ] H B H J B J. (6.4) Sn ( ) ( ) ( ) J J B (6.5) hfo af so aangn

151 ha 6. Dna Phonon sas n wo Dnsons 33 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) H J J H J J J J (6.6) whh s na o h souon n (4.3). A s souon Sn v fao n (6.) s nown fo ( ) a souon o h quaon shou b ossb. On ans of nua sovng hs quaon s o aaz an so ha Ζ (6.7) wh s a hosn funana fn fqun an s h s fqun souon. Equaon (6.) a b wn usng hs noaon as ( ) ( ) ( ) ( ) ( ) ( ) J B J H B H (6.8) wh n a h suaon wou b una o ov so s fqun ang. hs quaon an b wn n a noaon as W V (6.9) wh { } ( ) ( ) ( ) [ ] L L (6.3) { } ( ) ( ) ( ) [ ] L L (6.3) { } V V ( ) δ H B H V (6.3) an { } W W ( ) δ J B J W. (6.33) hus fo a una ang of fquns wh a fn fqun souon h as a fn an h ss s sovab: W V. (6.34) sng (6.34) w a fn a nw -vang -a fo sang fo on vang n as

152 ha 6. Dna Phonon sas n wo Dnsons 34 { n } { } [ V W δ ] (6.35) wh h δ n nas ha h saa na wavfunon os a ou baus w a ang wh a na sa. h a s n as { } n an osons o h sa f angua fqun n () h nn f angua fqun n () h sa f na wavfunon o () an h nn f na wavfunon o (n). B nng ou fnons of h sa an nn vos an h -a w an w h aonsh bwn sa an nn fs as (6.36) whh s h sa noaon as n h sa as. h n vos a shown n Fgu 6. an h n -a s shown n Fgu 6.. h ns of an an b aon an ons as bo vos. On h ag sa h bo oonns of h vo oson o h vaous oa haons an on h sa sa ah bo onsss of na wav os fo a aua haon. { } ( ) { } M M { ( )} Fgu 6.. h -vang sa f offn vo. h -vang oun vos of sa an nn f offns n (6.3) an (6.3) sv an b vsuaz as oun bo vos. h sa f offns a usa h bu h nn f offns a onfgu n an na ann. Shown f h ns on a ag sa oson o h vaous haons. Eah of hos ns s sf a oun vo of na wav os fo a aua haon. h -a an b nvson as a wo nsona bo a. In h ag a ah bo osons o h aonsh bwn sa an nn haons. Eah bo onans ns ha a h sa an nn na wav os. M M h a s n ha w us h haons fo bv n fn o su an ffn fquns wh s o h ung fqun.

153 ha 6. Dna Phonon sas n wo Dnsons 35 { } { ( )} O N N O { ( )} n O N N O Fgu 6.. A s -vang -a. On h ag sa shown abov ah n of h -a as sa f angua fqun wh nn f angua fqun. Eah n of hs a s sf a a ha as sa na wav o wh nn o n. sng h bo noaon h souon M O M N s N M O M. (6.37) Sn w a ang wh a n an w a usng a na ooson of h fs s agona n h n nsons. ha s s a bo a of agona as. n hs nw fnons h nn f s { } { ( )} n ( ) ψˆ ( ) wh ψ ˆ ( ) ψˆ ( ) ψˆ an ψ ( ) ψˆ ( ) sa f s { } ( ) u (6.38) { } ˆ a boh ow vos. Sa h s n ( ) ψ( ) wh ψ ( ) ψ ( ) { ψ } an ψ ( ) ψ ( ) u (6.39) { } a aso boh ow vos. h sa f fo a non-o -vang sa Whn h aa aas a vang n a non-o ann on anno s ons h quson n s of haons. h -a w s b s n h n nson bu w b a onnuous funon of h nn an sa f angua fquns an sv. hus

154 ha 6. Dna Phonon sas n wo Dnsons 36 ( ) ( ) ( ) (6.4) hs ngaon an b fo nua b onsng h -a n (6.35) an szng ( ) so ha N h nub of nn haons s suffn ag nough o san h nn su a a suab souon. h sa f fo a o -vang sa If h aa aa vaaon s o h onnuous aonsh n (6.) w gna no h quvan s aonsh n (6.9) sn a o su onsss of s oonns. W bgn b onsng a onohoa nn f of angua fqun an -vang aa aas of a aua ung angua fqun. h sa f w hfo onss of haons oa a whh s sa noaon us n (6.7) wh. h nn f s s sf b h offns ( ) ( ) δ an h sa f offns an b wn as h angua fqun-oan Fou ss ( ) ( ) δ (6.4) wh h usua fao of π has bn absob no h offns. Sa h nvs of h -vang ns of h n an s Fou ansfo a b wn as ( ) ( ) s s s P P ρ an ( ) ( ) s s s P P δ π. (6.4) Subsung (6.4) no (6.) gvs ( ) ( ) ( ) ( ) ( ) ( ) δ δ δ δ J B J H B H (6.43) ( ) ( ) ( ) ( ) ( ) ( ) J B J H B H δ δ δ δ (6.44) ( ) ( ) ( ) ( ) J B J H B H δ δ. (6.45)

155 ha 6. Dna Phonon sas n wo Dnsons 37 Subsung (6.4) no (6.9) gvs ( ) ( ) s s L s P B δ (6.46) wh ( ) ( ) ( ) J J L. (6.47) Subsung hs no (6.45) an sfng s ( ) ( ) ( ) ( ) s s s s s s J L P J H L P H δ δ δ δ. (6.48) aangng hs quaon on a -fqun bass b sovng fo gvs ( ) ( ) ( ) ( ) s s s s s s J L P J H L P H δ δ δ δ. (6.49) Baus of h a funons h fs suaon ov s s on nonzo whn s an h son whn s. Dszng h a funons wh h unsanng ha ah s oonn s h offn of a osonng a funon n h onnuous su J L P J H L P H δ δ. (6.5) As n h sz souon (6.5) an b wn n a a foa as W V (6.5) wh { } ( ) ( ) ( ) [ ] L L (6.5) { } V V H L P H V δ (6.53) an { } W W J L P J W δ. (6.54) No ha W s a oun vo sn h s on on nn fqun. hus W V. (6.55)

156 ha 6. Dna Phonon sas n wo Dnsons 38 W an o hs su o onfo wh h noaon n (6.36) n whh as wh [ V ] W n [ ] δ δ n fs o h h n of h V W vo. V W (6.56) If h nn f s no onohoa hn h souon w onss of a suaon of van haons fo a aua sa fqun. h van nn fquns a ( ). (6.57) onsng a h van nn fquns u- o own-oua o h fqun of ns h -a w b wh [ V ] W n [ V ] W δ n (6.58) s h h n of h V W vo aua wh. h n -a aonsh (6.36) a hn b u wh h nn offn vo. hs onus ou susson of sang fo a sng -vang n. N h u sang fo an suh -vang sas s ons an s shown o b v sa o h fouaon n sa as. 6.. Sang fo N oa -vang ns ons N oa -vang ns wh nown s -vang -as { } [ N ] wh an n a h sa an nn na wavs os n an an a h sa an nn angua fqun oonns sv. W wan o fn an quvan -vang -sua { } o o n so ha o o (6.59) wh w hav us h noaon nou n Fgu 6. an Fgu 6.. o us h a souons sn n Son w qu a nw -vang fnon of h saaon a. Essna ah haon s nvua ov usng h bo a h saaon a ns a S { S ( b )} wh ( b ) S ( b ) S n { } S. (6.6) ( b ) ψ n ( b ) δ n. (6.6) sng hs nw fnons w a s us h sa -a quaons fo u sas as n h sa as sa h fo onvnn:

157 ha 6. Dna Phonon sas n wo Dnsons 39 wh B δ I ( δ ) N N N o Sˆ [ S ]. (6.6) Sˆ. (6.63) h o of auang o If a h nvua -as a na hn h -sua a b a wh a ogah ouaona o wh s o h nub of sas. h agun as o wh hs s h as s ssna na o ha sn fo h D as n Son h -sua s aua fo wo hn fou hn gh. sas un h s -sua has bn a. h ou an o a aso na as n Son 5.4. ha a aaons of (6.63) a h ngaons n Fgu 5.5 an h aoanng susson. If h nvua -as a no na hn h -sua us b oos n a na ann. h n ho an b as aaz (uha) o un aoss u P os o on a P. h ns of h vaous as an b gna n aa an h ao of h anng oaons a a uaons an nvsons. Sn hs oaons an b ou n aa [3] hfo hs n ho w n sa ouaona gans as h FDD ho usng aa ouaon. 6. Vfaon h n ho fo oa -vang sas was oa agans anaogous FDD suaons fo boh sa an na ass. h suaon sus ah qu w as h subsqun fgus show. h onfguaon us n a h suaons was ns wh na ooons an os as n h honon sa sgn ha was suss n ha 3. h on ffn s ha h sffnsss of h sanss 33 w aus o sab ansvs wavs sn ou n ho os no aoun fo hs. o o hs w s an 44. h suaon gon was b 9 ( ẑ b ˆ ) wh s h n aus. h sou was oa 5.5 (8.638 ) n h - ẑ on fo h n of h suaon gon. In a h subsqun fgus of h san o vo f aus f vaus a noaz so h oou sa os fo bu o an sns vaus fo zo o on. o ou sus ha an b vsua oa wo saa FDD suaons w u o oban h nn an sa fs. h fs suaon onans on h sou an ovs h nn san f. h son suaon onans h sou an

158 ha 6. Dna Phonon sas n wo Dnsons 4 sas an ovs h oa f onssng of h nn us sa san fs. h nw V-MS ho auas h nn an sa fs as h vo ona whh s hn onv no h vo o oa wh h FDD sus. h FDD sus a sans an no vos bu sn h fs a oos b fqun an noaz h wo ss of sus an b oa. Fo bv w sha s f o boh fs as san fs. h FDD suaon us a aussan-aoz ansan sou. h nu us was aussan-oua wh a n fqun of 35 Hz an a Hz banwh nsa of a onoon whh ava h n fo sgna wnowng. h suaon gon onss of 56 ons n h ẑ on an 9 ons n h ˆ on. h s was 5 ns an h suaon uaon was 8.5 µs. hs was h uaon of wo fu nu wavfos. On sou of o n h FDD suaons aos baus h ansan sou naon has so afas sn a s gs. Fgu 6.3 shows h agnu of h san oonns aong h au an. h agnu of h sou saa sons n h ẑ on shown n bu os sbs h nn aussan sha fo aoa -6 o 6. Howv h a unnn sha us a hs ons. Aso h agnu of h sou saa sons n h ˆ on shown n shou b zo bu has a noab au n h suaon. A souon o hs obs was una bu h a on h ova sus s na. I s s aan ha h sus of h n ho os ah hos of h FDD ho..9 Noaz Sou Au u z u Au Pan () Fgu 6.3. h o FDD sou au on h au an. h ˆ an ẑ ons a aa an nua o h sou au an sv.

159 ha 6. Dna Phonon sas n wo Dnsons 4 o sua h aussan-aoz sou n h n ho h saa ansfo of h sou aozaon funon was us o gna an wavs of vaous angs an aus. hn ah an wav oonn was onv no na wavfunon offns usng h Jaob anson n (4.8). h offns fo ah o w hn su ogh o fn h s of nn wav offns n o onsu h sou. h w 5 wavvos us n h sou au ooson an 5 na wav os nu n h suaons. hs hos w suffn baus nasng h no su n an aab ovn n h qua of h suaon sus. Fou n ). haons us h funana w sov fo (.. [ ] 6.. On sa n V-MS FDD Fgu 6.4. h sua san fs fo a sng sa n. Pans a) hough ) show h nn oa an sa fs aqu fo h FDD suaons. Pans ) hough f) show h sus oban fo h n ho ab V-MS. h sgh saong vsb n h FDD suaons a as fo sou an absobng bouna onon nonas. As a fs h suaons of on sa n w fo usng boh h FDD ho an h nw -vang ho. I was shown anaa n Son 6.. ha h n ho oass o h sana u sang ho fo on n f h aa aas a onsan. Howv a h agans h w-sabsh FDD ho

160 ha 6. Dna Phonon sas n wo Dnsons 4 was s waan. h suaon sus showng h noaz agnu of h san au u z a 35 Hz a shown n Fgu 6.4 an sa n agn. W hfo onu ha ou n ho ous o sus fo a sng sa n. 6.. Sva sa ns A suaon of h sa ns a n a non-s onfguaon was sgn o vf ha h -sua fouaon s o n h sa as. Fgu 6.5 shows ha h wo ffn suaon hos ov h sa noaz u z san fs fo h sa ns. h a sgh sans bwn h sa an oa fs whh an b abu o h FDD suaon non-as bu h wo ss of suaons show n quaav agn ohws. V-MS FDD Fgu 6.5. h sua san fs fo h sa ns. Pans a) hough ) show h nn oa an sa fs aqu fo h FDD suaons. Pans ) hough f) show h sus oban fo h n ho ab V-MS. h s n quaav agn bwn h wo ss of fgus. h sgh sans a abuab o non-as n h FDD suaon On oa -vang n In hs s h fouaon fo h -vang -a fo a sng n was vf. h haas an of h n was vang snusoa b ±% a h

161 ha 6. Dna Phonon sas n wo Dnsons 43 vaaon fao. I was vang a a ung fqun of Hz. h vaaon fao was hosn so ha h fs wo haons abov an bow h funana fqun w ga han h nos foo of h FDD suaon. B nos foo w an h o nou hough sgna unaon u o a fn-uaon suaon f absobng bouna onons an h szaon of h suaon gon. Fgu 6.6 shows h noaz nn an oa f agnu sa of u z fo a on aong h ˆ nn an hafwa bwn h au an h n n. h nn f su s shown n bu wh h 35 Hz aussan nvo a vsb. h oa f whh s h su of h nn an sa fs s shown n. h sn of h fs an son haons a vn n h oa sgna. h nn us banwh an ung fqun w hosn o nz h ova of h nn us su wh h haons. Noaz Au (B) Fgu 6.6. h noaz san sa of u z fo a sng n. h on asu was aong h ˆ nn an hafwa bwn h au an h n n. h bu su s h nn us aon an h su s fo a na n wh vang b ±% a Hz. h fs wo haons a vsb n aon o h funana. h uvs a noaz b h nn su a 35 Hz.

162 ha 6. Dna Phonon sas n wo Dnsons 44 h noaz nn sa an oa u z san f agnus a 35 Hz a shown n Fgu 6.7 fo boh h FDD an nw V-MS hos. h sus ah qu w bu aa o ff v fo h sa as sn n Son 6... Howv on on ans h san f agnus a h haon fquns shown n Fgu 6.8 s a ha h n ho aua ahs h FDD ho un vang onons. h s n agn bwn h wo hos onsang h va of h nw V-MS fouaon fo a sng n. Pan ) n Fgu 6.8 shows so soon as oa o an h) bu hs an b abu o h oo sgna o nos ao a 55 Hz shown n Fgu 6.6. Inn Sa oa a) b) ) V-MS FDD ) ) f) Fgu 6.7. h sua funana fqun san fs fo on -vang n. h n an nu us a na o hos n Son 6.. ha h an of h n s vang snusoa b ±% a a fqun of Hz.

163 ha 6. Dna Phonon sas n wo Dnsons 45 V-MS FDD Fgu 6.8. h sua san f haons fo on -vang n. h n s h sa as shown n Fgu Sva oa -vang ns In hs fna vfaon s h fouaon fo h -vang -sua was vf fo h ns. h ns a na o h on n Son 6..3 nung h vang os. Fgu 6.9 shows h noaz nn an oa f agnu sa of u z a h sa on hosn fo Fgu 6.6. I nas ha h ho of a vaaon fao of % agan as h fs wo haons suffn abov h nos foo ha h a b a fo h FDD suaon sus. h agnu of h oa f su a 35 Hz s u u o suv nfn bwn h nn an sa fs a h hosn asun oaon.

164 ha 6. Dna Phonon sas n wo Dnsons 46 Noaz Au (B) Fgu 6.9. h noaz san sa of u z fo h ns. h on asu was h sa as n Fgu 6.6. h bu su s h nn us aon an h su s h oa f fo h na ns. Agan h fs wo haons a vsb n aon o h funana fqun. h uvs a noaz b h nn su a 35 Hz. h funana fqun fs a shown n Fgu 6. fo boh h FDD an n hos. Agan h sus ah v w fo h funana fqun. As n h sa as h a sgh sans bwn h sa an oa fs whh an b abu o h FDD suaon non-as. h san fs a haon fquns a sn n Fgu 6. an onf ha h n ho aua ahs h FDD ho fo o han on -vang n. h a aso so sgh sans bwn h wo hos bu h a u o h FDD suaon non-as; n aua h sou au. Ohws h s n quaav agn bwn h wo ss of suaon sus.

165 ha 6. Dna Phonon sas n wo Dnsons 47 V-MS FDD Fgu 6.. h sua funana fqun san fs fo h -vang ns. V-MS FDD Fgu 6.. h sua san f haons fo h -vang ns. h ns a h sa as shown n Fgu 6..