HOLOGRAPHIC FABRICATION OF WOODPILE TYPE PHOTONIC CRYSTAL TEMPLATES BY THE USE OF ONE-DIMENSIONAL DIFFRACTIVE OPTICAL ELEMENTS. Zsolt L.

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1 HOLORAPHIC FABRICATION OF WOODPIL TYP PHOTONIC CRYSTAL TMPLATS BY TH US OF ON-DIMNSIONAL DIFFRACTIV OPTICAL LMNTS b Zslt L. Pl BS. In lcticl ngining, Univsit f Pittsbugh, 005 Submittd t th dut Fcult f Swnsn Schl f ngining in ptil fulfillmnt f th quimnts f th dg f Mst f Scinc in lcticl ngining Univsit f Pittsbugh 009

2 UNIVRSITY OF PITTSBURH SWANSON SCHOOL OF NINRIN This thsis ws psntd b Zslt L. Pl It ws dfndd n Nv. 0 th, 009 nd ppvd b Jl Fl, PhD, Pfss Willim Stnchin, PhD, Pfss nd Chimn ungng Li, PhD, Assistnt Pfss Thsis Advis: Kvin P. Chn, PhD, Asscit Pfss ii

3 Cpight b Zslt L. Pl 009 iii

4 ABSTRACT HOLORAPHIC FABRICATION OF WOODPIL TYP PHOTONIC CRYSTAL TMPLATS BY TH US OF ON-DIMNSIONAL DIFFRACTIV OPTICAL LMNTS Zslt L. Pl, M.S Univsit f Pittsbugh, 009 As th sch f fbictin tchniqus twds th pductin f lg dfct f thdimnsinl phtnic cstls cntinus, hlgphic lithgph psnts itslf s pssibl slutin. In this thsis, simplifid mthd tht is f f cmpl pticl stups is dmnsttd. Within th c f th mthd psntd lis dil vilbl pticl cmpnnt, phs gting tht b dsign psnts gin f intfnc vilbl f lithgphic pcsg. Th phs gting hibiting n-dimnsinl pidic ngmnt dsignd t diffct int th substntil ds ncssitts tw psus ft which thdimnsinl pidic ngmnt is lid. Th ngtiv tn pht sist, SU-8 utilid t cd th dsignd intnsit distibutin pvs itslf s vibl intmdi twds high dilctic cntst stuctus. Th pviusl stblishd lg bndgp phtnic cstls psnt fbictin chllngs nd thus ppimtins t ths stuctus hv bn ppsd. Th spcific mthd mpld pns th d t nl n f th pviusl stblishd chmpin phtnic cstls but nvthlss th mst sught ft dimnd stuctu pdictd t hibit n f th lgst pssibl bnd gps. Th wdpil stuctu psssg sm f th qulitis f th dimnd iv

5 lttic is pvn t b n dqut pcticl ppimtin nd nc ppl dsignd hibit lg bnd gps. Th spcific tchniqu mpld pmits th pltin f th FCC spc gups lng with th FCT nd Ttgnl spc gups. Th fscintin tht ths stuctus hv pvd is fuld b th vst pdictd pplictins ncmpsg nl ll nwn scintific disciplins. On ds nt hv t vntu f t li th ptntil hld f th tlcmmunictin indust such s dns wvlngth multipls, high fficinc lss, lss f pviusl unvilbl wvlngths, sup cntinuum sucs, flt lnss, suppisms, lsslss wvguids, nd snnt cvitis t mntin fw. Dvlpmnts f ths dvics wuld pgss th dvncmnt f tchnlgis such s pticl stg, dug dliv sstms, nd dvncd imging. Sm hv vn cmpd th discv f ths mtils t th vlutin chivd b th smicnduct indust with th dvnt f cntllbl lctnic bnd gps. v

6 TABL OF CONTNTS ACKNOWLDMNTS... X.0 INTRODUCTION....0 BACKROUND INFORMATION PB THORY AND CHARACTRISTICS MAXWLL S QUATIONS IN PRIODIC STRUCTURS Fui pnsin f th Pmittivit NRATION OF TH INTRFRNC PATTRN Intnsit Distibutin Dsign....4 CRYSTALLORAPHIC ANALYSIS SOLVIN TH WAV QUATION DRIVIN CHMISTRY XPRIMNTAL PROCDURS XPRIMNT STUP FABRICATION PROCDUR XPRIMNTAL RSULTS STANDARD VALUATION DIFFRACTION ANALYSIS ANALYSIS... 6 vi

7 5. SUMMARY FUTUR WORK APPNDIX A. UNION OPRATION FOR OBJCTS APPNDIX B. MATLAB COD THAT PRDICTS TH OMTRY OF TH STRUCTUR... 7 APPNDIX C. MATLAB COD THAT SOLVS FOR TH DISPRSION RLATION APPNDIX D. SAMPL MPB COD USD FOR TH PRDICTION OF TH DISPRSION RLATIONS... 8 BIBLIORAPHY vii

8 LIST OF FIURS Figu. Dispsin ltin mpl, bnd digm f unifm slb, b bnd digm f D pidic stuctu Figu. Bnd pning dmnsttin b slutin dcupling... 6 Figu. Ointtin f -vcts f light diffctd b th phs gting... 9 Figu 4. Nmlid intnsit distibutin dpicting th intfnc pttn gntd b th phs gting... Figu 5. Nmlid intnsit distibutin f tw thgnl psus... Figu 6. Digm dpicting th tw tms in th pimitiv cll lng th spctiv inttins... 6 Figu 7. vlutin f th fist Billuin n with chngs in pidicit tis... 8 Figu 8. Fist Billuin n f wdpil stuctu with C/L /... 9 Figu 9. Th fist Billuin n f 60 stuctu lft, with tp viw t shw th spctiv ngul disttin ight Figu 0. Bnd stuctus f wdpil ngmnt with C/L, dii C/8. -th bnd stuctu b Mtlb with clindicl tms, b-b MPB with clindicl tms, c-b MPB with ctngul tms, nd d-fist Billuin n with lbld -vcts Figu. Th vitins in bndgp with chngs in pidicit tis f th tw psu ttins Figu. A gnl schm f phtsnsitid ctinic plmitin. Rdwn fm Rf Figu. Digm f th light intnsit psnt in th gin f intfnc with sptd cmpnnts int DC nd scillting tms. Rdwn fm Rf viii

9 Figu 4. pimntl stup usd t psu smpls t dsignd intfnc... 5 Figu 5. Sufc SM img f smpl hving 60 stuctu Figu 6. Sidviw SM img lft shwing ptil pning, Sufc pticl micscp img ight shwing cllnt sufc stuctu Figu 7. SM img f 60 stuctu, lft cls mintin f th sufc, ight mintin f th sid Figu 8. SM img shwing full pnd cstl with sufc viw ight nd sid viw lft Figu 9. Sufc img shwing lng ng d tp lft, cls viw with pdictd stuctu inst tp ight, sid viw with pdictin inst bttm lft, tp viw f nth cstl pln bttm ight Figu 0. Diffctin img tn f 90 stuctu with HN ls lft nd whit light ight... 6 Figu. Diffctin img tn f 60 stuctu with HN ls lft nd whit light ight... 6 Figu. Diffctin img tn f 45 stuctu with HN ls lft nd whit light ight... 6 i

10 ACKNOWLDMNTS Th cntinuus finncil nd cdmic suppt civd fm D. Kvin Chn md this sch pssibl nd I wuld b miss if I did nt cgni his ncugmnt. I wuld li t cnwldg th suppt I civd fm D. Jl Fl b llwing th us f his quipmnt nd lbt f this pimnt lng with ssistnc in thticl nlsis. Sm dvitins tn with th pimnt ld t th us f D. Albt Hbl s quipmnt nd ssistnc f which I m gtful. Th hlp I civd fm D. Stvn Jcbs, s h hlpd with pblm I hv ncuntd in dvlping th Fui tnsfms, llwd m t stblish cmplt nlticl dsciptin f th ts t hnd f which I m gtful. Th suppt I hv civd fm Ms. Snd Wisbg in fm f pfssinl dvic nd psnl ncugmnt lng with ssistnc nsuing tht I civd finncil id in fm f fllwships nd ssistntships ws pmunt thughut m gdut c. I wuld li t thn th Univsit f Pittsbugh f pviding m with Tching Assistntships f th dutin f m gdut stud. I wuld b miss if I did nt mntin th hlp I civd fm m fu lb mts. Di Xu s thticl nd pimntl ssistnc ld t th succss f th pjct. M discussins with Michl Buic, Chuc Jwt, nd Bn McMilln nd thi ssistnc in nlsis w intgl t m succss. Lstl, I wuld li t thn m fmil nd finds f lws ncuging m t succd. Withut thi suppt, nn f this wuld hv bn pssibl.

11 .0 INTRODUCTION Th stud f lctmgntic wvs in dilctic mtils in pidic ngmnt bgn v cntu g. Th bhvi f lctmgntic wvs in ths pculi mtils ws fist mind b Ld Rligh in 887 in cnnctin with th stng flctiv pptis f ctin cstllin minl tht hs pidic plns t which twinning ccu. H cncludd tht ths stuctus hibit nw ng f fquncis phibiting th ppgtin f lctmgntic ditin nd tht this gp is dpndnt n th ngl t which th light bms ntd th mdi. Th ving pidicit pincd b ditin impinging t nnthgnl incidnc ld t his bsvtin f flctd cls highl dpndnt n this ngl. A mdn mintin f ths stuctus gups thm int th fmili D phtnic pstls lng with thi uniqu pptis. Fllwing his w v th nt cntu, cnsidbl fft hs bn givn t th stud f multil films nd hs bn put t us in mn pplictins nging fm ntiflctin ctings t nhncing th fficinc f LDs t highl flctiv mis in ls cvitis VCSL. It ws nt until 987 whn S. Jhn nd. Yblnvitch, b thi mging f lctmgntism nd slid stt phsics, shd light n th cncpt f mnidictinl bnd gps in tw nd th dimnsins., Ths nvl ngmnts gntd gt intst nd hncfth mn dvlpmnts hv bn md twds th fbictin nd pplictin f ths

12 mtils, lt tmd phtnic cstls. Du t th difficult f fbictin in th pticl gim, l ws w cnstind t thticl studis nd stuctus in th micwv gim. In 996, T. Kuss dmnsttd th fist tw-dimnsinl phtnic cstl t ninfd wvlngths b bwing fbictin mthds usd in th smicnduct indust. 4 Dspit th succss f ths dvlpmnts, th fist tw-dimnsinl phtnic cstl tht fund cmmcil intst ws in th fm f phtnic cstl fib, dvlpd b P. Russll in 998. Th wll stblishd fbictin mthds in th smicnduct indust pditd th dvlpmnts f tw-dimnsinl phtnic cstls hwv this ws nt th cs f th thdimnsinl cuntpt. Th lc f vilbl tchniqus t l n ld t slw pgssin in dvlpmnts f th-dimnsinl cstl nd n ttmpt ws md b bwing isting mthds vilbl in th smicnduct indust lng with sm clv thining which ld t th l-b-l fbictd wdpil stuctu. 5,6 Th hustin f th vilbl tchniqus nd th difficult tht ccmpnid thm ld t th dvncmnt f nw nd ctiv pcsss ncmpsg cllidl slf-ssmbl 7, multiphtn dict ls witing 8, nd multibm hlgphic lithgph 9-6. Amng th vilbl tchniqus twds th fbictin f th-dimnsinl phtnic cstls, hlgphic lithgph hs ttctd th mst ttntin f its inhnt flibilit. Althugh its mplmnt pducs intmdi tmplts b th us f infilttin nd invsin tchniqus, 7 ths cn b cnvtd int high dilctic cntst stuctus. Tditinll, multi bm intfnc lithgph ws lid b lg numb f bul pticl cmpnnts, such s mis, bm splitts, nd lnss. 0, Ths pticl stups inhit mn dgs f fdm dmnding pcisin in lignmnt nd suscptibl t thml vitins nd mchnicl vibtins. Rcntl, numb f gups hv dmnsttd th cnstuctin f multi bm intfnc

13 ug gl dflctiv diffctiv pticl lmnts -5 lng with lithgph bsd n multipl diffctiv lmnts n n glss ms nd gl flt tp pism. B mpling tw thgnll lignd phs mss, phtnic cstls with wdpil stuctus hv bn cntl dmnsttd thugh n tw ls psus. 4-6 Cntinuus ttmpts t ducing th cmpnnts quid ith b utiliing m cmpl lmnts b th mging f multipl lmnts int gl cmpnnts sd th bttl with lignmnt stmming fm thml vitins nd mchnicl vibtins nd phs mtching quimnts f multipl vlpping bms. Ths dvncmnts nt f fm th cnstictins tht ccmpn thm s ths pticl stups psssg ducd dgs f fdm ls limitd t hlfspc. Tht is, n sid is usd f th gntin f th intfnc nd th th is f th cding f th intfnc, thus sulting in lngtin lng th ptic is ccmpnid b ducd cstl smmt nd bnd gps. Th w psntd h dmnstts th fsibilit f pducing lg dfct f phtnic cstl tmplts ug diffctiv pticl lmnts s plcmnt t bul pticl stups lng with n ttmpt t gin lst cstl smmt du t lngtin.

14 .0 BACKROUND INFORMATION. PB THORY AND CHARACTRISTICS Phtnic cstls, stuctus cmpsd f ltnting high nd lw ind f fctin, gnll but nt limitd t dilctics, ngd in sm pidic schm thid nd hv bn dmnsttd t hibit uniqu chctistics tht psnt nwfund cntl v lctmgntic ditin. Anlgus t its lctnic cuntpt, tms mlculs ngd in pting schm psnt pidic ptntil t ppgting lctn. F lctns, th mtil nd th gmtic ngmnt dictl ffct th cnductin pptis f th cstl. lctns tvg cnducting cstls withut sctting psd pblm tht th phsics f th ws nt quippd t hndl, but this pculi bhvi ws plind b th dvnt f th th f quntum mchnics. This nw phsics ttd th lctns s wvs, nd wvs tht mt spcific citin culd tvs thugh pidic ptntil withut sctting; cnvsl, th stuctu culd hibit n ng gp tht phibits th ppgtin f ths wvs in ng f fquncis. Phtnic cstls in which th cnstitunts plcd b mcscpic mdi with diffing dilctic cntst nd th pidic ptntil is plcd b pidic dilctic functin cn psnt mn f th sm phnmn t phtns s smicnducts d t lctns. 4

15 Ths nvl tificil mtils cn pvid ct cntl v th ppgtin f lctmgntic ditin. Onc dsignd t fulfill ctin dilctic cntst quimnt, th cn phibit th ppgtin f light in ng f fquncis lng ctin dictin ll dictins. Fbiddn gps in Phtnic cstls cn b ith mnidictinl ptil, ncmpsg ll pssibl dictins f ppgtin just subst f wvvcts. Rgdlss f th spn f th gp, th igin f th pning cn b undstd b mining ndimnsinl pidic sstm. A slb f dilctic with unifm dilctic cnstnt hs pln wv ignslutins ωc fd t s th light cn. This sstm is spcil cs f pidic stuctus wh pid f ptitin ppchs ilding th unbundd dispsin ltin. Lbling th stts in tms f Blch nvlp functins nd wvvcts f sm >0, vls th dgnc inhnt in this sstm. Th wvvcts, >π/ fldd int th fist Billuin n dshd lin in figu, th - π/ md lis t n quivlnt wvvct t th π/ md with th sm fqunc nd s n. Figu. Dispsin ltin mpl, bnd digm f unifm slb, b bnd digm f D pidic stuctu. 5

16 A dict w f bsving th igin f th pning f fbiddn gps is ccmplishd b dcmpg th wv slutins int lin cmbintins f s nd cs nd distubing th unifm dilctic functin with pidic cmpnnt whth th distubnc is usidl εδcsπ/, sm th scillt functin. Figu. Bnd pning dmnsttin b slutin dcupling Th psnc f n scillting dilctic distubnc bs sm f th pviusl stblishd dgnc b fcing th c slutins t ccup high ε gins nd th slutins t lw ε gins. This phs shifting btwn th tw slutins is dictl spnsibl f th pning f fbiddn gps. Appling ptubtin th und th cnstint f smll vitin shws th bndgp s functin f mid-gp fqunc t b Δω/ω Δ/. Fm this it fllws tht n pidic mdultin f th dilctic functin sults in th fmtin f fbiddn gp whil smll mdultin sults in smll gp nd vic vs. Th fmtins f cmplt bndgps in high dimnsins qui m pcis ngmnts. Alng ch pth tn th will b n pning f fbiddn gp pdictd b th D gumnt. Th fmtin f glbl gp is ccmplishd b nsuing tht ll th D 6

17 gps vlp in fqunc nd tht ch n is sufficintl lg. Alng ths n-dimnsinl pths th mid-gp fqunc, s it invsl lts t th pidicit cπ/ ε ½, lds t th smmt quimnt f simil pidicitis in ll dictins. Hncfth, th tingul lttic in tw-dimnsins nd th fc cntd cubic lttic FCC in th-dimnsin psss th lgst gps. mining hw th filds bhv t th bundis vls t nth citin. At dilctic bund, whth th lctic fild is plll ppndicul, th ng dnsit ε will b discntinuus css th intfc stmming fm th bund cnditins tht nd ε cntinuus. Th discntinuit f th ng dnsit givs is t difficultis in cnfinmnt f th ppndicul lctic fild lins in th high ε gins nd th plll fild lins in lw ε gins. Fcting in this ltt quimnt, th dsignd stuctu shuld b cmpsd f thin cntinuus chnnls unning in ll dictins, fcilitting stng cnfinmnt f lw bnds nd th fcing ut f th high bnds, s ths thin shps cnnt suppt multipl mds. 8 Th wv qutin dscibing lctmgntic ditin in dilctics is f f pmts cnsticting th slutins t spcific lngth scls, und th ssumptin f mcscpic mdi, nding th wv qutins invint f scl. This is v usful ppt s difficult t mnufctu stuctus t th pticl lngths culd b mnufctud t lg, m sil ttinbl scls nd th msud chctistics with sm lin djustmnts dscib th stuctus t ll lngth scls. This lds t simpl ltinships btwn lctmgntic pblms tht diff nl b th lin scling f distncs nd mplituds. If th dilctic functin is mdifid b linl djusting th spcing ε/p, wh psnts th dilctic functin s sptil dpndnc nd p is th scling fct, i.. cntctin 7

18 pnsin. Aft psg thugh th wv qutins, th scling djusts th fqunc ignslutins b ω/p nd th md slutins b H H /p. O, if multipl slutins dsid f th sm stuctu t diffnt dilctic cnstnts, linl scling th pmittivit mplitud ε/p nd psg it thugh th wv qutins shws tht chnging th mplitud f th dilctic cntst hs n ffct n th fild distibutins nd it simpl scls th fqunc ignslutins b p / ω. 8 Anth intsting ppt tht ds nt qui lg dilctic cntsts, nd f tht mtt vn significnt pning f fbiddn gins, is th nmlus fctin bsvd t th intfc f phtnic cstl with tht f hmgnus mdium. Und ctin cnditins, th fctin ngl is fund t b v snsitiv t chngs in th incidnc ngl nd wvlngth. This chctistic is ttibutd t th nistp f th bnds nd th dispsin tht lctmgntic ditin culd pinc is thid t b ds f mgnitud gt thn tht bsvd in cnvntinl pisms; hnc, its pppit nming cnvntin, suppism. Sinc its intductin b Ks t l., 9 mn hv dmnsttd this intsting ppt in tw-dimnsins nd lt in th-dimnsins. Jsp Sbin nd Min u cntl dmnsttd tht suppism phnmn cn t plc in plmic wdpil stuctus with n ind cntst f.6 lw, nd pidicit ti f.8. Th pidicit ti is dfind s th pid f ptitin f th stuctu in th stcing dictin dividd b th pidicit in th mining dictins. Th stng dispsin chctistics f phtnic cstls thught t significntl dvnc wvlngth divisin multipling tchnlgis mngst ths. Ks t l. hv ls dmnsttd tht phtnic cstls cn hibit lns li bhvi fund in cnvntinl ptics. 0 Th hv bsvd tht slf cllimtin mngst th lns li bhvi 8

19 cn t plc in th-dimnsinl phtnic cstl, bhvis tht tpicll is fm nn lin ffcts K ffct cug slf fcug slitn ppgtin quiing high ng inputs nd cn v sil cllps. In phtnic cstls, th fund t b indpndnt f th light intnsit nd th quimnts tpicll mpld t pvnt bm divgnc diffctin bdning.. MAXWLL S QUATIONS IN PRIODIC STRUCTURS Phtnic cstls, tificil stuctus dsignd pimil f th mnipultin f lctmgntic filds, ntull mind b th qutins tht dscib ths filds. Fu qutins, uss s lw, uss s lw f mgntism, Fd s lw f inductin, nd Amp s lw with Mwll s cctin gvn ll lctmgntic filds gdlss f fm nd, thf, ncmpss th filds in ths stuctus. ρ ε H 0 H t H µ J µε t uss s lw lts th lctic chg cntind within clsd sufc t th suunding lctic fild. uss s lw f mgntism fmults tht th ttl mgntic flu thugh sufc is ; tht is t s, tht mgntic chgs cm in pis dismisg th istnc f mgntic mnpls. Fd s lw f inductin cupld with Amp s cicuit lw dscibs hw chnging mgntic fild is lws ccmpnid b n lctic fild nd vic vs. 9

20 Mwll s dditin f th displcmnt cunt t Amp s lw cmpltd th th b dding chnging lctic filds t th mi. Bf Mwll s dditin, it ws blivd tht mgntic filds gntd nl b lctic cunts. Th iginl fm f th displcmnt cunt dscibd plitin cunt in dilctic mdium nd ld Mwll t div th wv qutin b ssciting this displcmnt cunt t tim-ving lctic filds. A mdn ppch tht plcs th iginl divtin nd svs sm cumbsm mthmtics cmbins Amp s cicuit lw with Fd s lw f inductin. Th divtin bgins b th mintin f Mwll s qutins in th Hvisid fm with th fllwing cnditins. Fist, th mdium is suc f nd lsslss, tht n cunt ists, nd tht ε is l. Nt, th mdium is tim-invint such tht n md cn b dcmpsd int st f pln wvs b Fui th. And finll, tht th pmbilit insid th mtil is cnstnt. Ting th cul f th tw ltins givs th fllwing: H µε t t t H H µ µ ε t t t µ 0ε 0, µ µ 0µ, ε ε 0ε c Ug th css pduct vct idntit nd th ssumptin tht th filds tim-hmnic, th wv qutins btind in th fqunc dmin. w µ ε c w H µ H ε c 0

21 Th wv qutins in diffntil fm pvid gd stting pint f n mthd chsn t slv pticul sstm. Th tw bsic ppchs in slving th wv qutins in pidic mdi th Finit Diffnc Tim Dmin Mthd FDTD nd th Pln Wv Mthd PWM. Bth mthds with sm dditinl w cn ld t th sm sults, but th FDTD is pimil usd f finding th fild distibutins s functin f tim within stuctu whs th PWM lds dictl t th dispsin ltin. Bth hvil l n dns cmputtinl dt in btining high dgs f ccuc nd, thf, bth cmputtinll intnsiv. Fst mthds hv bn dvlpd tht cn dsticll duc cmputtin tim nd dt stg quimnts but th PWM is dvlpd nd implmntd s stting pint. ith wv qutin, whth it is functin f H, cn b usd t slv th sstm, but in pctic th wv qutin tht is functin f H is pfd. Th H fild dpndnt wv qutin is ld in th fm f psitiv dfinit Hmitin ign sstm; hwv, with sm dditinl w th fild dpndnt qutin cn b cst in tht fm s wll. Th dpth f th cmplictin in chg th fild wv qutin is chd whn impg th tnsvslit cnstint s it dpnds n th pmittivit. If instd th lctic displcmnt fild is chsn, th dpndnc n th pmittivit f th tnsvslit cnstint is limintd, but thn th pmittivit pps th tims in th wv qutin nd will undubtdl cmplict th cmputtin, s it is nw, is in th fm f gnlid ignsstm. Th H fild dpndnt wv qutin is ld in m plibl fm nd tnsvslit cnstint f th mgntic fild is sil impsd. Th H fild insid th stuctu cn b wittn in tms f ppgting functin multiplid b functin with th sm pidicit s th stuctu:

22 R h h h H i Th pidic functin h is Fui pndd int th sum f pln wvs nd cmbind t fm i i h H h h Impg th tnsvslit cnstint which ssntill stts tht mgntic mnpls d nt ist givs 0 0 i h h H Th mgntic fild is nml t th dictin f ppgtin nd lds dictl t th sptin f fild cmpnnts int thi spctiv dimnsins, 0 wh,,, unit vcts nd fm n thnml tid llwing th dcmpsitin f th mgntic filds int thi dictinl cmpnnts.,,, ν ν h h h h v Th dcmpsd mgntic fild cn nw b intducd int th wv qutin, qutin b ting its cul.

23 [ ] [ ] [ ] i i i h i h i H nd h i H ~ ~ ν ν ν ν ν ν ν ν ν κ κ κ H, th Fui pnsin f th invs f th pmittivit is intducd, i ~ κ ε κ Th vibls nd th vibls f th pnsins f th mgntic fild nd th pmittivit nd ducd b quting. Ting th nt cul, w gt [ ] [ ] i h H ~ ν ν ν κ κ Rcmbining th wv qutin with th dcmpsd mgntic fild nd th pplid culs, w gt [ ] ~ i i h c w h ν ν ν ν ν ν µ κ Th wv qutin cn b pushd futh b dividing ut simil pnntil fcts, subtcting fm th pnntil fcts n bth sids, ming us f th dlt functin idntit b intgting v ll spc, nd substituting f c it is dumm vibls. d i V δ π B intgting v ll spc nd b ug th Dic dlt-functin idntit, [ ] ~ h c w h δ π µ δ π κ ν ν ν ν ν ν

24 4 Fm bv, substituting f nd ug th fllwing idntitis, i j ij i ij nd j i j i δ δ δ 0 th wv qutin bcms [ ] ~ ν ν ν ν ν ν µ κ h c w h wh, th qutin is multiplid b v nd is simplifid b ming us f th fllwing vct idntitis [ ] [ ] [ ] ν ν ν ν A C B B A C C B A Appling th tnsfmtins, t, t nd t, th ltin will b in th fm f pg. 9 [ ] [ ] ~ h c w h ν ν ν ν ν µ κ A m usful fm f this ltin fm th pspctiv f cmputtin is in mti nttin. Th tms cn b wittn in th fm f vct mgnitud multiplid b th dictin unit vcts. [ ] [ ] [ ] [ ] ν ν ν ν [ ] [ ] ν ν Th ltin wittn in ign vlu fm,

25 ν ψ ν, ν w h µ v h c v ψ ν, ν ~ κ 4 Th ltins ivd t dil implmntd b n cmputtinl tl such s Mtlb nd th vctil cmputtinl ids. Th vibls nd fund b clsl mining th stuctu in cipcl spc wh th vcts th dictins chsn in th Billuin n nd th vcts th cipcl lttic vcts. An ccut psnttin f th invs f th dilctic functin κ is pmunt nd dictl dtmins th ccuc f th sults, but cn b cumbsm t btin... Fui pnsin f th Pmittivit Th difficult in slving th ignvlus f th ignsstm lis in btining n ccut psnttin f th pimitiv cll in th Fui dmin. Finding th nlticl tnsfm f th pimitiv cll will gtl duc cmputtin si nd tim, but n nlticl tnsfm is nt ncss nd smtims vn t difficult t btin nd in such css n wuld hv t l n Fst Fui Tnsfm FFT lib nd th pblms tht ccmpn thm. Sm f ths lig nd cnsidbl lg mtics tht in tun qui m stg spc nd cmputtin tim nd sults in pblms with cnvgnc. 4 Fui th dictts tht n pidic signl cn b cnstuctd t its iginl fm b th suppsitin f n infinit numb f pln wvs. In pctic snbl dg f ccuc cn b chivd with 5

26 finit numb f pln-wvs. Th dilctic functin f th stuctu bing itslf pidic is wittn s sum f pln wvs. ε i Ω ε Ω 5 wh ε is th pmittivit, is Rcipcl lttic vct, nd Ω is th vlum f th lmnt cll. Th pimitiv cll cntins tw llipticl clinds displcd nd ttd but th igin nd cn vlp n nth. Th dict ttmnt f llipticl clinds ffst nd ttd fm nd but th igin is cumbsm; thf, utiliing th pptis f th Fui tnsfm is idl. Th llipticit, th ffst, nd th ttin dlt with b th us f Fui tnsfm pptis tbultd b Bcwll nd tmd ffin tnsfms. 5 Cicul smmt simplifis th fm f th pmittivit in th spctiv dimnsins t cicul stp functin nd in th mining dimnsin th clinds t th fm f squ wvs. ε ε ε ε ε ε b b R b R h R h R Rc p p Rc p p Rp Rp 6 h, th intductin f nd p is f th simplicit f illusttin s th ncmpss th ttin nd th llipticit, nd th displcmnt is shwn b p nd p. Th tw pmittivit smbls, ε nd ε b psnt th dilctic cnstnts f th ds nd th bcgund spctivl. Th Π smbl is usd t psnt th stp functin, Rc is th dius f th ds, h is th lngth f th ds. In css wh th stp functin is functin f th dius, th stp functin is cnsidd s cic functin. Th tnsltin vct Rp llws th pmittivit functin t spn v th whl spc. T ccunt f th pssibilit f vlp, in such cs whn simpl dditin f th tw clinds pducs nus sults, n ptin tht mgs th tw clinds whil tining 6

27 7 th iginl hights is dvlpd. Th unin ptin cnnt b pplid dictl but cncptull, it dscibs th ptin th wll. In css wh n vlp ists nd cnnt b vidd th mthmticl quivlnt f th unin hs t b pplid nd its dvlpmnt is psntd in Appndi A. T bgin th Fui ttmnt f th pimitiv cll, th qutin f n llips is cnsidd s stting pint. In clculus whn dling with th intgtin f n llips, it is cmmn pctic t pfm chng f vibls tht pull ut th llipticit s multiplicnd nd intgt v th mining unit cicl. A gnl qutin f n llips is b 7 wh, is th dius f th mj s, nd b is th dius f th min s, chnging th vibls nd t nd givs, b, Th stndd w f finding th intgtin fct ccmpning chng f vibl is b th us f th Jcbin, dfind s 6 pg. 974 v u v u v u v u v u wh dudv v u v u h v u g f da f R S,,,,,,,, 8 F th tnsfmtin pfmd bv t th llips, th intgtin fct is

28 , b 00 b, Th qutin f th Fui tnsfm f th pmittivit hs th pnntil multiplicnd tht hs l spc nd Fui spc dimnsin vibls which must b tnsfmd s wll. Rplcing, nd with thi quivlnts, cllcting tms, nd dtmining th tnsfm f th Fui spc vcts, w iv t b,, b i i i Th vibl chng hs ducd th cmplit f intgtin du t llipticit nd nw vibl chng t duc th cmplit f ttin is dvlpd. Bcus th clinds ttd but th -is, nl th vibls psnting th -pln will chng, quivlnt t pfming tw-dimnsinl ttin. Th stndd sstm f qutins tht tun th nw vibls ft tw-dimnsinl cuntclcwis ttin is 7, cs β β β cs β 9 Appling th ttin tnsfmtin ducs th cmplit f intgtin b ligning th cdint s with tht f th clind. Substituting in th ttin qutins nd thn slving f nd ncssitts slving th sstm f qutins. 8

29 β cs β cs β β β cs β β β cs β β cs β cs β β cs β β cs β 0 cs β β β cs β Fllwing th sm mthd n m tim, th qutin f is fund. cs β β cs β β β cs β cs β cs β cs β β β β β cs β cs β β cs β β cs β β An tim vibl tnsfmtin is dn th intgtin fct ccmpning th sid tnsfmtin nds t b fund. Ug th Jcbin, th intgtin fct is fund t b, cs β cs β β β, As pctd, th intgtin fct ducs t c ttin tnsfmtin will b bsbd b th bunds v which th functin is intgtd. As bf, th vibls in th pnntil tm nd t b mdifid t cmplt th chngs. Substituting f nd with th nwl fund quivlnts 0 cs β β β cs β nd b cllcting tms nd ssigning vibls, w iv t th tnsfmd pnntil tm 9

30 cs β β β cs β cs β β β cs β i i i Th ncss tnsfmtins tht simplif thwis cmpl mthmtics hv bn stblishd. This is llwd bcus th sid tnsfmtins lin nd blng t clss f ptins tmd ffin tnsfms. Th sulting ltins dvlpd in Appndi A dling with th vlp nw intducd. Th, th pmittivit functin is wittn s n lgbic sum f its individul cmpnnts, th tw clinds with hight f ε sitting in bcgund f ε b. Substituting this pmittivit functin int pmittivit intgl nd bing up th intgl, w gt ε Ω Ω Ω ε b Ω i [ ε ε ε C C C C ] b ε ε b d Ω b Ω C i d Ω C i d i d Ω C C i Dfining th intgl f th pduct f C nd C s th vlp intgl, nd ug th Fui tnsfm dulit, tht multiplictin in l-spc is quivlnt t cnvlutin in th Fui dmin, th intgl bcms I vlp Ω Ω Ω C Ω i C C d Ω Spting th st f th intgls f s f hndling, i Ω C d i d d 0

31 I Ω I I Ω Ω i ε b Ω Ω Ω C C d i i d Ω d Ω Ω R Ω R h R h R Rc p p Rc p p Rp Rp Sinc th fist intgl is simpl th Fui tnsfm f cnstnt, it ists in th Fui dmin s dlt functin, thus it will nl influnc th cfficints t th cnt f th i i spctum, i.., ll cipcl lttic vcts f lngth. And it simpl vluts t I Ω ε b Ω i ε b 0 d 0 lswh Th vlutin f th st f th intgls quis littl m w. Th summtins in th intgls cn b dppd c nd p spn th whl dmin nd ths cupld with th tnsltin vcts R nd Rp, spn th whl spc. I b i h R Rc p p Rp d Ω Ω Th slutins cn b dcupld int tw gins whn th cipcl lttic vcts ist nl t th igin nd whn th ist lswh. Cnsiding th cs f th igin whn is, b I h R Rc p p Rp d Ω Ω This intgl vluts t ilding th fctinl vlum f th clind with spct t th vlum f th spc it sits in, d d I b Ω h Rcπ d bhπrc p dp dθ Ω h 0 0 Nt, th cs whn th cipcl lttic vct,, is th thn is mind. Th stp functins mvd fm th intgl s ths bsbd b th limits f intgtin.

32 I b Ω b Ω ip p ip p h h i h R d Rcπ 0 0 p i ip p dp d Rc p p Th dt pduct in th pnntil tm is cnvtd t sphicl cdints, Rp ip p dp ip p i p, p cs p θ, p cs θ Substituting f nd nd f nd, w gt p p p p [ θ cs cs θ ] θ p p cs cs θ nd b ming us f th pduct t sm tignmtic idntitis, th cipcl lttic vct is wittn s, p p p p cs θ Ming ths substitutins in th intgl, w iv t I h Rc π b ip p i d p dp Ω h 0 0 ip p csθ Th fllwing idntitis lt th f pidic pnntil functin in n pid t fist d Bssl functin, nd llw simplifictins in th intgls. d

33 π J π 0 0 ipp csθ α d J α d J p p 0 α Ths tnsfmtins llw I t b wittn nd slvd s I h Rc πb ip p i d Ω h 0 πb Ω ip p πbh c Ω i i h h h Rc J p p J p p J 0 p p p Rc p p dp Rc 0 ip p Th vlutin f intgl I is dn much th sm w s it ws dn f I. Th fm f th qutin is th sm, nd th nl chngs tht ctull ccu th vlus f th tnsltin nd ttin. Th sults btind f th vius intgls ll hv cnstnts tht cmmn btwn thm nd ths pulld tgth int gl dscipt, th fctinl vlum. bhπrc f Ω At this pint, th sults cn b cst in ith fm, th dict ttmnt f clinds with vlp b pcding with n ltnt lmnt cll tht pvids th sm sults withut hving t cnsid vlp. Th dict ttmnt tht llws f vlp f th clinds in th lmnt cll vls th pmittivit functin in Fui spc. mining th slutins b cnsiding th cipcl lttic vcts t nd t th thn lds t th finl fm f th pmittivit functin. Whn th cipcl lttic vcts th pmittivit ds nt dpnd n vibls nd its vlu dfins th hight f th spctum t. Cllcting ll th sults t givs

34 ε 0 I 0 ε ε ε b ε 0 ε ε ε b b ε ε b I 0 I0 I 0 I0 [ f f f f] b [ f f ] Whn th cipcl lttic vcts cqui vlus th thn, th pmittivit in cipcl spc is dfind b th pduct f fist d Bssl functin with tht f Sinc functin. I, f Sinc[ β, ip β h J[ p β, Rc] ] p β Rc,, p, Rclling th tnsfmtins tht w pfmd nd substituting bc t,, nd tms, th vibls β cs β β cs β,,,, p cs β β,, b Th numicl subscipts f th tnsltin p nd ttin β spcif th clind tht is cnsidd within th pimitiv cll. Th finl fm f th pmittivit in Fui spc tht cmbins th slutins in ll th dmins nd llws th clinds in th pimitiv cll t vlp, is ε b ε ε b f f ε ε ε b 0 I I I I lswh 0 If n ltnt pimitiv cll is fund tht ds nt cntin vlps f clinds, thn ltin dscibing tht cs cn b fmd b ducing th btind sults. Th djustmnt ncmpsss nl th mvl f vlp tms f ll slutins. At, th hight f th spctum t is ducd b mving th dditinl tms cquid b th vlp. [ f f] ε 0 ε ε ε f I 0 I 0 ε 0 I0 ε ε b ε ε ε b b b b 4

35 At cipcl lttic vcts th thn, th pmittivit functin is mdifid b mving th cnvlutin tm fm th qutin with vlp. Th finl fm f th pmittivit f th cs f nn-vlpping clinds in th Fui dmin is ε b f ε ε b ε ε ε b I I 0 lswh. NRATION OF TH INTRFRNC PATTRN Th lctmgntic ntu f light llws us t plit its inhnt pidicit t ngin multitud f sptill pidic intnsit distibutins. Ths distibutins cn b cnvtd int spcil clss f nw mtil, phtnic cstls, vi hlgphic lithgph. Mnufctuing phtnic cstls b th cding f dsignd intfnc pttns pvids cntl v mn dgs f fdm which cn b mnipultd t pvid stuctus f diffnt cstllgphic clsss. Hlgphic lithgph invlvs th cding f fmtins f sptill pidic light intnsit distibutins fm th intfnc f tw m chnt light bms. Th lctic fild f pln mnchmtic light is mthmticll dscibd b i m m wtm m,, m t wh m is th ind f th bm, is th wv vct, w is th ngul fqunc, th cnstnt is th lctic fild stngth, nd is th initil phs f th bm. Th psnc f multipl bms intd such tht th vlp in sm gin f spc pduc nw distibutin, nd this distibutin is simpl th vct sum f th individul cmpnnts. Th intnsit in th gin f vlp f th multipl bms is pptinl t th squ f th sultnt vct sum. 5

36 6 Th mthmticl dsciptin shwn is gnl n nd it pplis t ll tps f plitin f mnchmtic light whth lin, cicul, llipticl. Sinc th tmpl cmpnnt in th dsciptin fl ppgts withut n lttins, it flls ut f th intnsit qutin du t th inn pduct with its cmpl cnjugt. n m n m m m m m m m m m m m m m i N m N n n m i n i m wt i n wt i m I * *,, *,, * * * H, N is th numb f pln mnchmtic bms f light tht vlp nd th th tms s dscibd bf. F linl plid lctmgntic ditin, th intnsit qutin cn b wittn in simpl fm 8 < N n m n m n m n m N m i I ] cs[ δ δ And f th cs wh N is, th intnsit qutin simplifis t ] cs[ I This is th qutin f n-dimnsinl intnsit distibutin in th wds, ndimnsinl gting with sptil pid f As shwn, th intfnc f tw bms pducs n-dimnsinl pidic intnsit vitin whs th intfnc f th nd fu bms pducs tw nd th dimnsinl intnsit vitins spctivl. Sinc n f th mn pincipls f ngining is t gnli difficult scintific dvlpmnts int sil pducibl nd cst ffctiv mthds, th

37 gntin f th intnsit vitin is pducd with n ff th shlf phs gting Ibsn Phtnics. Th phs gting is dsignd such tht it hs sptil pid f 900nm nd it diffcts int nl th ds f substntil intnsit, th 0 th nd th ± st ds with n fficinc f 97.%. Of this intnsit 6% is diffctd t th 0 th d nd 5.% t th ± st ds. Th lctic filds f th th bms diffctd b th phs gting mthmticll dscibd b, t, t, t,,, i i i wt wt wt Ths th bms thn intf in th gin th vlp t pduc th pidic intnsit distibutin cdd b th pht sist. Th gin f vlp is dtmind b th css sctinl f th impinging bm nd th ngl f diffctin. Sinc th bms pduc tw-dimnsinl pidic intnsit vitin nd th gl is t hv th-dimnsinl vitin, th pht sist is subjctd t tw psus. Th scnd intnsit vitin is just mdifictin f th fist such tht th sultnt stuctu hibits th-dimnsinl vitin. Th tw mthds vilbl tht mdif th scnd st f bms t chiv th dimnsinl intnsit vitin. Th initil phs cmpnnts in th qutins ncmpss this dg f fdm. Sinc th initil phs in hlgphic lithgph is gnll bit nd will nl chng th stting vlu, it cn b ignd f th fist psu but it will bcm pmunt f th scnd psu. Th phs cmpnnts in th intnsit qutin f th scnd psu cn b cntlld b ith phsicl displcmnt f th stting psitin b plcing phs shifting plt in pth f th bm. ith mthd pducs th sm sult which is phs diffnc btwn th fist nd scnd psus. Th ttl Intnsit tht th pht sist is subjctd t is th sum f th tw intnsit distibutins. 7

38 8 < < ] cs[ ] cs[, j i i j i j i j i j i i j i j i j i i i ttl t I δ δ δ δ 4 Wittn ut int ll f its cmpnnts with th phs cmpnnts plcd b displcmnts in th dictin, th ltin bcms ] cs cs cs cs cs[ ] cs cs cs cs cs[ ] cs cs cs cs cs[ ] cs cs cs cs cs[ ] cs cs cs cs cs[ ] cs cs cs cs cs[ ttl I θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ A fid wvlngth ls impinging n th phs ms will mg s th bms s dpictd in th fllwing digm. H, th s bitil chsn nd ngd such tht th bm ppgts in th psitiv dictin. Th digm dpicts th -vcts t sm ngl tht nd phi in sphicl cdints. Tht is th ngl f ttin f th bms whil phi is th ngl diffctin f th ls bm insid th pht sist.

39 Figu. Ointtin f -vcts f light diffctd b th phs gting Th ngl phi is nt uniqu mngst th diffctd bms, th cntl bms pssss n vlu f phi nd th ths pssss nth t nml incidnc. As mntind bf, phs gting is usd t pduc th in pln intfing bms t sm ngl. This ngl is fund b ug Snll s lw insid th pht sist nd is fund t b 9.8⁰ whn th wvlngth f th ls is 54.5nm in f spc nd pplis t ll nn-cntl bms. Th th bms pducd b th phs gting hv tw diffnt lctic fild intnsitis; th cntl undictd bm cntins 6% nd th tw diffctd ns cntin 5.%. This is th sm f th scnd psu s wll. Appling th fmntind simplifictins, gnl fmul tht dscibs th intnsit vitin s functin f th ngl f ttin, diffctin, nd displcmnt is 9

40 0 ] cs cs[ ] cs cs cs[ ] cs cs cs[ ] cs cs[ ] cs cs cs[ ] cs cs cs[ 4 I ttl θ θ θ θ θ θ θ θ θ θ θ θ Tht ts n tw vlus, n bing th ngl f ttin f th fist psu, nd th scnd bing th ngl f ttin f th scnd psu. Phi ts n n vlu nd is th ngl f diffctin. As tw ttin ngls dmnsttd, tw intnsit qutins shwn dscibing th spctiv intnsit distibutins. Th spcific intnsit distibutin f th thgnl psu is pt smmtic in th dictins t mi th inttin f th stuctu mind lt. ] cs[ ] cs cs[ ] cs cs[ ] cs[ ] cs cs[ ] cs cs[ 4 90 I Th intnsit vitin ts n diffnt fm s th ngl f ttin is vid. Th scnd vitin dmnsttd is whn th ttin ngl tht ts n th vlu f 60⁰.

41 ] cs[ ] cs cs[ ] cs cs[ ] cs[ ] cs cs[ ] cs cs[ 4 60 I.. Intnsit Distibutin Dsign Sinc th intnsit vitins thn cdd b th pht sist, th phs dl fcd n th scnd st f bms hs t b such tht th ttl intnsit vitin is bth th-dimnsinl nd wll intcnnctd. Th intcnnctivit is cucil spct nd it will dtmin whth th pcssd stuctu will suppt itslf nd min cstllin. Th intfnc f th fist th bms pducs tw-dimnsinl intnsit vitin in th fm f llipticl clinds ngd in tingul.

42 Figu 4. Nmlid intnsit distibutin dpicting th intfnc pttn gntd b th phs gting Th digm shws th qul ng cntus f nmlid intnsit f 0.6 nd bv; hwv, ng is psnt in th nti gin but it is f lw vlu thn 0.6 nd s f th pups f illusttin it is mittd. Th ng f th bcgund is n tifct f th intfnc nd is cntibutd t mstl b th cnstnt tms in th vll intnsit qutin. This bcgund ng is n imptnt tpic tht will b dscibd lt. Subsqunt psu plcs scnd tingul f clindicl intnsit vitins displcd fm th fist b sm distnc such tht th ls full intcnnct.

43 Figu 5. Nmlid intnsit distibutin f tw thgnl psus Th pidicit f th cnstuctd intnsit vitin is dtmind b mining th simplifid intnsit qutin f th tw sts f bms. Th pids f ptitin invint f tnsltin in th -dictin nd th fund b stting th spctiv cmpnnts t nd ting th ptil divtivs in th dictins cnsidd. F th cs f thgnl psu, th pidicitis f th stuctu in th nd dictins d nt diff.

44 4 L I,, 90 cs 0 cs 4 0 cs cs 4 λ λ λ π π π π 5 Th pidicit f th intnsit vitin in th -dictin is ls invint f tnsltin nd is fund b ting th ptil divtiv in tht dictin whil hlding th nd dictins cnstnt t n bit vlu. Th finl fm f th pidicit is shwn s functin f wvlngth nd ngl f diffctin insid th pht sist. λ λ π π cs cs cs cs 4 60 C I 6 Fllwing th sm pcdu n m tim vls th pidicit f th stuctu cmpsd f intnsit vitins ttd b 60 dgs fm n nth. Th invinc f th pidicit in th stcing dictin t ttin is bvius s it is nt functin f th ngl f ttin. λ λ λ π π,,, L 7

45 Th munt f phs shift fcd n th scnd psu b th tnsltin f th igin dtmins whth th pcssd stuctu is wll intcnnctd nd is cstllin. Th quid tnsltin in th -dictin t chiv cmplt nd smmtic intcnnctivit is sil btind b mining th vlppd intnsit vitins nd is fund t b qut f th pidicit in th -dictin. 0 4 λ 4[ cs ] 8 With this tnsltin th ls f clindicll cnfind ngis will b intcnnctd with n nth t fm th-dimnsinl vitin in intnsit which nc pcssd will fm th cstl tmplt. Finll, n qutin llwing th pltins f vius cntst tis s functins f th tw ngls is vld. C L λ cs λ θ θ θ cs 9.4 CRYSTALLORAPHIC ANALYSIS Th nlsis f th pidic ntu f th intnsit vitin is dn thugh wll stblishd st f tls in slid stt phsics. It is stblishd tht th-dimnsinl cstllin stuctus cn v btwn 7 cstl sstms nd 4 Bvis lttics. 9 It is imptnt t nw tht hw mn f ths 4 Bvis lttics th tmplt cn b cst int s sm f th intfnc pmts vid. This infmtin will undubtdl simplif n futh nlsis nd psnts n with cl viw f th undling pptis f n stuctu. Aft cful 5

46 pltin, it is fund tht th stuctu f th tmplt cn b cst int mdifid dimnd stuctu with tw-tm bsis. Th dimnd stuctu with tw tm bsis is gnll fd s th incblnd stuctu nd is pt f th cubic cstl sstm nd it is fc cntd Bvis lttic FCC. Hwv, th tmplt hibits cubic smmt nl und ctin cnditins nd thf in th nlsis, th pidicit in th stcing dictin, - dictin, must b llwd t v. F C/L / th lttic cn b cnsidd s n FCC pimitiv unit cll nd t th tis it is fc cntd ttgnl FCT pimitiv unit cll. T cst th stuctu f th tmplt int th incblnd fm th stuctu in Ctsin cdints is lignd such tht th nd s lng th [0] nd th [-0] dictins whil th cdint is mins unchngd. 0 This lignmnt is pcis f n psu ttin f 90⁰; t ths, th lignmnt hs t b djustd. Figu 6. Digm dpicting th tw tms in th pimitiv cll lng th spctiv inttins With n thgnl psu, smmtic vcts tht dscib th stuctu th FCC pimitiv vcts. Th id is t find bsic st f vcts tht ft intgl ptitin will pduc th stuctu withut n vlp gps. 6

47 7 0 Cmputtins in -dimnsins v lg s cumbsm with spct t tim nd stg quimnts. It is pmunt t duc pimitiv cll t nn-pting gin minimiing th tnt v which clcultins pfmd. Th cntinutin f th nlsis in cipcl spc bings fth imptnt ftus such s Bg plns nd minimum nnpting gin in which cmputtins lss cumbsm. Th tnsfmtin f n pidic lttic t cipcl spc is ccmplishd b th fllwing th pimitiv vcts tht dpnd nl n th l spc lttic vcts. 9 b b b π π π This llws th cnvsin f th l spc lttic vcts int cipcl spc, nd ths b b b π π π Th -spc Wign-Sit pimitiv cll, ls nwn s th fist Billuin n, is th gin in which st f pints ists tht cn b chd fm th igin withut csg n Bgg plns. Th fist Billuin n cn still b cmputtinll lg. Finding st f pints

48 within tht tul minimi th vlum f cmputtin is dn thugh smmt ptins tht ild gin tmd th iducibl Billuin n. 8 Th fist Billuin n f n FCC cstl lttic is th bd cntd cubic Wign-Sit cll, but s th ti f th pidicit vis, th gmt f th Wign-Sit cll ls vis with this chng. Figu 7. vlutin f th fist Billuin n with chngs in pidicit tis As th C/L ti vis fm.707 t 5 th gmt f th fist Billuin n spns fm BCC nd FCC hbid t highl disttd FCC, n FCT Billuin n. With C/L / ti th fist Billuin n cquis full FCC smmt. 0 Within th fist Billuin n, minimum gin ists tht is tul nn-pting nd is fund b mining th undling smmt f th pimitiv unit cll. Th iducibl Billuin n f th cs f full FCC smmt is th vlum gin within th citicl pints. 8

49 9 Figu 8. Fist Billuin n f wdpil stuctu with C/L / F sphicl tms th unpimd citicl pints sufficint t dscib th iducibl Billuin n, but f th cs f clinds th dditinl pints quid du t bn smmt ptins. Th scnd cs pld is whn th ttin f th scnd psu is t 60⁰ ngl t th fist. In this cs th smmtic st f pimitiv vcts is fund b mining th stuctu nd th vcts f thgnl ttin in cnjunctin with th nw pimitiv cll. Pimitiv vcts tht cctl pduc th 60⁰ stuctu in l spc nd cipcl spc b b b π π π Th fist Billuin n is fund in much th sm w s it ws f th thgnl psu: th lttic pints mind in cipcl spc b idntifing Bgg plns tht fm th

50 40 bund f th stuctu. Th smmt f th Billuin n flcts th nn-thgnl psu ttin sn in Figu 9 ight. Figu 9. Th fist Billuin n f 60 stuctu lft, with tp viw t shw th spctiv ngul disttin ight.5 SOLVIN TH WAV QUATION Th wv qutin dscibing th intctin f lctmgntic ditin with tht f pidic stuctus is divd nd plcd int vctil fm. Th pssin in such fm simplifis th implmnttin int vilbl cmputtinl ids, f mpl Mtlb, f slving th ignvlus f sstm., v v h c w h µ ψ ν ν ν, ~ κ ψ ν ν

51 isting in th fqunc dmin, th qutin dpnds n vibls tht dscib th stuctu in cipcl spc. Th vibl is minimid st f cipcl lttic vcts lng th dgs f th fist Billuin n. Th vct st,, cnsists f intgl cmbintins f th cipcl lttic vcts dscibing th numb nd lctin f tms in th vicinit f th igin. m b mb mb Th unit vcts i dscib th tw dictins nd fund b mining th dictins f. Th pmittivit f th stuctu ncmpsg ll th gmtic dtils is psntd b κ in th ltin. Hving divd n nlticl qutin f th stuctu in th fqunc dmin, th duus ts f ug Fst Fui Tnsfm FFT lib with its ccmpning hudls is vidd. Th fdm f psu ttin llws nn-thgnlit in th pimitiv cll nd s cnsqunc, finding n ccut Fui psnttin b mns f FFT wuld pv t b chllnging. Alng with th sid difficultis tht is fm nn thgnl sstms, lig nd th tifcts tht pllut th sults wuld ccmpn n FFT pcg. Althugh finding th nlticl tnsfm f pimitiv cll culd pv t b chllnging s it ws f th cunt stuctu, t th mmnt it is pfbl ut in cmpisn t th ltntiv. Th ltin dscibing th sstm is Hmitin ignvlu pblm nd cn b implmntd b numicl mthds. Just s Fui th dictts, n pidic signl cn b pfctl cnstuctd b n infinit numb f pln wvs; hwv, in pctic snbl ccuc is chivd b finit numb f thm. Sinc th pblm ist in th-dimnsins, cnsiding nl smll numb f pln wvs, n, lng ch dictin plds th pblm int Nn mti. Nding nl tw dimnsins, th th dimnsinl mtics tnsfmd int tw-dimnsinl ns fming mtics f NN si. Fm ths it fllws tht it is pfbl t hv pimitiv cll cmpsd f smthl ving bjcts s th qui 4

52 fw pln wvs t pduc thm with th sm ccuc s wuld bjct with shp dgs. Th pimitiv cll f th wdpil stuctu gntd b th mthd mpld in this thsis is lngtd in th stcing dictin ilding highl llipticl clinds s tms in th pimitiv cll. This nti-smmt quiing high fqunc cmpnnts f ccut pductin in th Fui dmin dmnd m pln wvs. Th nd f high numb f pln wvs is ls ppnt upn mining th clinds lng thi s s in ths dimnsins th psntd b squ wvs. Impvmnts tht plc discntinuus functins with smthl ving ns b sm st f vging much ndd s cmputtins in th cunt fm will fst bcm tim-cnsuming f stndd dstp cmputs. Sm pptis f such sstms dscibd b wv qutins simplif th implmnttin f cmpnnts b cnsiding nl ctin pts f th pblm. Th pt ψ is lin tht is if h nd h slutins, thn hh slutins. Th pt is Hmitin, f,ψg ψf, g. Th dilctic functin is psitiv dfinit ilding l ignvlus. Th gnl schm f implmnttin cn b cmpundd int fw stps. Fist, th cipcl lttic vcts clcultd fm which th vct st is fund f chsn numb f pln wvs. Th th dimnsinl mtics dscibing psitin in th Fqunc dmin thn tnsfmd int tw-dimnsinl ns ft which th pmittivit mti is slvd. Nt, th pdtmind -pints lng which clcultins pfmd input nd this st is futh pndd b lin intpltin. Lstl, th unit vcts fund b mining th dictins f in th spctiv dimnsins. Th ignvlus slvd b Mtlb s builtin functin lng with pp tunctins. Th lg si f th mtics tuns vst numb f 4

53 ignvlus; hwv, nl th lwst d mds f imptnc nd thus th tund ignvlus tunctd t 0 mds. Th Mtlb pgm tht slvs f th dispsin ltin is fund in Appndi C. Th ignvlus tund b th cunt implmnttin vlidtd with publishd ws 0 nd with MIT s Phtnic Bnds Pcg MPB. Th dimnd stuctu cmpsd f sphicl tms pvd t b vibl tst stuctu s th smthl ving ftus fcilittd smll cmputtins. Th hundd nd ft th 4 pln wvs tund indiscnibl sults fm tht fund b MPB. Hwv, th wdpil ngmnt cmpsd f clindicl tms pss cmputtinl difficultis n Pntium 4 dstp cmput with b f m. T tst th Mtlb implmnttin, smmtic wdpil ngmnt is picd with C/L f nd tmic dii f C/8 intd thgnll. 4

54 Figu 0. Bnd stuctus f wdpil ngmnt with C/L, dii C/8. -th bnd stuctu b Mtlb with clindicl tms, b-b MPB with clindicl tms, c-b MPB with ctngul tms, nd d- Fist Billuin n with lbld -vcts. Th plts f Figu 0 dscib th dispsin ltin lng with th lbld pints in th fist Billuin n. Th Mtlb implmnttin pducd th bnd digm with 75 pln wvs ching th limit f th sid cmput nd cnsuming 8. hus f th cmputtin. Th dict implmnttin f th pln wv mthd with plllid cmputing cpbl f hnding v lg cmputtins will pduc v ccut sults, but its unvilbilit ld t th us f MPB t supplmnt nd pfm th mn cmputtins quid t mp ut th sstm. Whil MPB is bsd n th pln wv mthd, it is implmntd b ming us f vging mthds t smth ut discntinuitis in th dilctic 44

55 functin nd b ug ittiv ignslvs in slving f th ignvlus which in tun gtl duc th cmputtinl sucs ndd. Th bnd digm gntd b Mtlb hs lmst cnvgd t th sults btind b MPB b; th slight vitins th sult f t fw pln wvs in th Fui psnttin. Th nd t pfm clcultins t vius C/L tis ld t th mintin f ug ctngul tms s MPB ds nt hv th cpbilit t implmnt llipticl clinds. Th dispsin ltins fund b squ tms simil t ths fund ug clindicl ns lbit n vll fqunc ffst. Ug ctngul tms will pvid gd ppimtin f mining th si f th bndgp t vius C/L tis nd spcill t gt lngtins. Th pidicit ti, C/L ws dfind s th ti f th pidicit f th stuctu in th stcing dictin t th mgnitud f th pidicit lng th th tw dictins. Th smpl MPB cd tht clcults th dispsins ltin whil mimiing th bndgp with spct t th ltl tmic dimnsin is fund in Appndi D. 45

56 Figu. Th vitins in bndgp with chngs in pidicit tis f th tw psu ttins..6 DRIVIN CHMISTRY At th bs f hlgphic lithgph cmpl chmist ists nbling th cding f dsignd intnsit distibutins t fm dubl stuctus withstnding th stsss f lithgphic pcsg nd f vius invsin schms. Th cmmcill vilbl SU-8 mitu Micchm cnsisting f divtiv with til sulphnium slts s phtcid gnts disslvd in gnic slvnts such s cclpntnn is suitbl ngtiv tn pht sist f th pductin f high slutin plmic stuctus. SU-8, dsignd f th fmtin f multipl films with fin ftus, is usd t pvid films up t mm thic with 46

57 cntst ti f 0 gt. Anth quimnt tht SU-8 mts is tht it is highl tnspnt in th n-uv nd visibl gins pmting unifm psus thughut th film. Th intnsit distibutin is gntd b ls in th visibl gim nd c SU-8 is fmultd t b snsitiv t UV, it ds nt spnd significntl t visibl light; hnc, sm mdifictins md. Th dditivs tht nbl th cding f visibl intfnc pttns phtsnsitis tht bsb visibl light nd tnsf th chgs t th nium slts thugh th fmtin f chg tnsf cmpls. Th phtsnsitis usd 5,7-diid--but- 6-flun nd phnl-p-ctlphnl-idniumhfluntimnt H-Nu 470 nd OPPI fm Spct up Limitd, Inc.. Upn psu t UV light, th phtsnsiti tnsfs th cquid chgs t th sulphnium slts gnting phtcids in lclid gins. Th high tmptu insnsitivit f SU-8 pls pivtl l in minimiing cid diffusin bf pst psu b nd it pvnts plmitin duing psu mng th nnlin chngs. 47

58 Figu. A gnl schm f phtsnsitid ctinic plmitin. Rdwn fm Rf. Thml cittin subsqunt t psu cclts cid diffusin nd inducs ctinic plmitin. Th undpsd sist is thn mvd b ppln glcl mthl th ctt PMA fm Micchm fllwd b cful mvl f slvnts b isppl lchl IPA nd ding. Th ding pcss is citicl s stuctus with fin ftus cn cllps fm sufc tnsins btwn liquid nd i intfcs nd it is fund tht ug IPA cn gtl duc wht wuld thwis b sufc tnsins if wt is mpld instd. Th DC tm in th intnsit distibutin lng with th lw ng scillt cmpnnts cmplicts things s th ls gnt phtcids sulting in nn- cidic bcgund tht blus th sptin btwn psd nd undpsd gins. If nt cnsidd, this bluing will pvnt th full pning f ps dding cmplictins t subsqunt stgs f th pcss. 48

59 Figu. Digm f th light intnsit psnt in th gin f intfnc with sptd cmpnnts int DC nd scillting tms. Rdwn fm Rf. Th nn- bcgund culd b minimid b cntlling sm f th tms in th intnsit qutin, b ptimiing th lding f th phtsnsiti, b chmicl mns. Th distibutins f phtcids dtmin th gins tht plmi nd dding n cid scvng ducs th cncnttin nd minimis cid diffusin. An pppit munt f Tithlmin TA which ptill nutlis th phtcids gntd b th nn- bcgund is mpld s mthd f cntl v th ffct f th bcgund ng. 49

60 .0 XPRIMNTAL PROCDURS. XPRIMNT STUP Th gntin f th intfnc pttn quis svl cmpnnts tht dil vilbl. Th ls bm is pvidd b n Agn In ls Spct-Phsics Mdl 00 djustd f mnchmtic utput t 54.5nm. Th utput f th ls is linl plid nd th ccmpning lctic fild scillts in th hintl pln. Nt, th ppgting bm intcts with n lctic shutt du t th smll it ptu f th shutt nd it llws pcis cntl v th psu tim dwn t 00ms. Th ctiv ls bm is thn nlgd. This nlgmnt is ncss s it dtmins th lngth f th gin v which th th-mging bms vlp pducing usful intfnc. Th ls bm is nlgd b bicnv tw-lns sstm t bm dimt f but hlf n inch utiliing th vilbl gting. Th bm itslf cnsists f svl mds nd th dditinl mds th thn th TM 00 distt th intfnc pttn b ducing th cntst btwn th mimums nd minimums in th scillt cmpnnts s wll s dding t th DC tms incg th cnstnt bcgund ng. Th mvl f th dditinl mds is ccmplishd b µm dimt sptil filt plcd t th fcl pint f th mgnifictin sstm. This pticul si f th sptil filt is 50

61 clcultd t b ptiml t th wvlngth f th ls, th incming bm dimt, nd th pptis f th lns usd t fcus. Figu 4. pimntl stup usd t psu smpls t dsignd intfnc Nt, th mgnifid nd cllimtd ls bm intcts with th phs gting diffcting it int th substntil ds. Oiginll, th phs gting ws plcd nt ttin stg t llw ttin but th ptic is; hwv, this pvd t b undsibl s it ddd unwntd displcmnts btwn th fist nd scnd psus. T mv th undsibl dgs f fdm fm th phs gting psitining, tw phs gtings fid nt plishd qut wf t th chsn ngl f ttin but th ptic s. Th smpl hld tht psitins th ppd pht sist plll t th phs gting is muntd n high pcisin mtin stg Nwpt PM500-4L tht llws displcmnt in th nd dictins with ±00nm pcisin. Th intfnc stup in its ntit lng with th Agn In ls is muntd n vibtin dmping pticl tbl nd fltd n i pcts thugh th psu pcss. Th pw pvidd b th ls bm is msud t b 0.45W immditl bf impinging n th phs gting. 5

62 Th pht sist usd t cd th dsignd intfnc pttn is SU-8 05, which is high spct ti ngtiv tn p bs plm. Its mimum bsptin is f ultvilt light t 65nm nd hncfth it is mdifid t tnd its bsbnc int th 55 nm ng. SU- 8 is mdifid b dding phtsnsiti t bing th intctin wvlngth int th 55nm gim, c-initit cting s ctlst, nd ppln cbnt PC t ssist in th disslutin. Th phtsnsiti is 5,7-diid--but-6-flun HNU-470 nd hs bsbnc f 0.5 in th wvlngth f intst. Th c initit, 4-ctlphnl idnium hflu-ntimnt OPPI, ssists in th chg tnsf f th chmicl ctin subsqunt t psu. A smll munt f pht sist is ppd t tim, sufficint f but 0 ds f tils du t th snsitivit f th mitu t thml nd lctmgntic ditin. Th pht sist slutin is cmpsd f 40g f SU-8 05, 0.5wt% f H-Nu-470,.5wt% f OPPI, nd 0ml f PC. Th mitu is stid f svl hus t nsu cmplt disslutin tht, in tun, sults in high cntst stuctus. Thugh pimnttin, th difficult f btining th dsid sult bcm tnspnt nd is thught t b cusd b th lg bcgund ng psnt in th intfnc pttn. T s this hudl, it is fund tht th dditin f Tithlmin nutlis sm f th Lwis cids gntd b psu f th pht sist t ditin. Th gntin f Lwis cids is pmunt s it llws th cding f th intnsit pttn; hwv, th bcgund ng psnt ls gnts ths cids dtimntl t th pcss b ducing th cntst btwn th psd nd undpsd gins. Th nlsis f th smpls is dn b tw tps f imging nd simpl diffctin tst. Onc smpl is thughl d it is mind b high slutin pticl micscp with cmput imging intfc. mintin b n pticl micscp llws ugh sting f 5

63 smpls int tw ctgis: ths tht culd pssss th dsid stuctul sults nd ths tht d nt. Th smpls dmd f futh nlsis thn clsl mind b Scnning lctn Micscp SM tht is cpbl f slving ftus in ds f mgnitud gt thn dsignd f this tmplt. Finll, if n tw smpls min in th btch f dns, th subjctd t diffctin tsts tht qulittivl vl th disd psnt n lg scl. A cln diffctin img ws th sult f lg dfct minimum gin. Oth smpls tht hd ith v ugh sufcs smll gins f stuctul intgit hd diffctin imgs with lw intnsit ps suundd b sufc scttd light spts. Th svl vibls tht pl l in th pcss, sm intndd nd sm nt, ch within its wn fm f fnc qui lg numb f ittins t chiv cnvgnc. Th intndd st f vibls p nd pst b tims, spin cting spd, th tim lngth f psu, th tim lngth f dvlping, nd th wight pcnt f TA with spct t th munt f SU-8. Oth vibls dtimntl t th pcss stm fm th pcisin quid t lign th tw sts f intnsit distibutin ccding t dsign citi. Th displcmnt f th phs gting b mtin stg td t ±00nm with ±00nm unctint cntibuts significnt munt f with spct t th quid tnsltin, spcificll in th vicinit wh th tw intnsit clinds vlp s th f th mtin stg h is cmptivl significnt. Th lignmnt f th pticl stup quis th mst ttntin, s it is pmunt tht th tnsltin f th scnd psu ts plc lng th ptic s nd hving n quntittiv mthd implmntd t cntl this vibl, it pss th gtst unctint. Th p nd pst 5

64 bs dh t stblishd pcdus dvlpd b svl gups;, sm vitins w pld withut n significnt dvitins. Th vibls f spin cting tht dtmin th thicnss nd th unifmit f th SU-8 sufc hld cnstnt t pviding thicnss f but 0µm. Th thicnss f th SU-8 mitu spun nt th substt cn, f cus, v thughut th pcss f fbictin du t th vibl cntnts f TA mng ths. Cntl v th vibilit f tim lngth f dvlping is td b dvlping v smpl f sht tim pids, nd mining ch n und lw light cnditins nd pting until hustin. Th mplmnt f ths stps ducd th vibls quid f cnsidtin t th tim lngth f psu, tnsltin, nd TA cncnttin. Th quid tnsltin is nwn vibl; hwv, displcmnts btwn th tw gtings muntd n qut wf culd s hv unwntd ffsts tht cupld b th f tnsltin dmndd ttntin. Th tw vibls pld b hlding tnsltin cnstnt t n incmntl vlu t which th psu tim is vid linl in pdtmind ng. Th pvius mthd is mpld f ch slutin mitu f SU-8 cntining vius munts f TA fm -40 ml%.. FABRICATION PROCDUR Th pcss f pducing th tmplt cn b split int th stgs. In th fist stg, th smpl is ppd f psu; nt, th smpl is psd; nd, thid, th smpl is pst pcssd t vl th dsignd tmplt. In th smpl pptin stg, th substts plishd glss slids ttd with pinh slutin nd dhumidifid b bing n ht plt. ch substt is ctd with thin l f OmniCt µm t nhnc th dhsin f SU-8 t th substt. Th pht sist mitu is spun nt th ttd substts t spds f 700 t 500 pm, 54

65 sulting in ng f thicnsss fm 5 t 5µm. Th ppd smpls w thn pbd t 65⁰C f but 0 min t mv th slvnt fm th pht sist mitu. Th lngth f th pb ds nt qui pcisin s its gl is t mv th slvnt nd thf it cn b subjct t lttins. High tmptu pb is nt dvisd s it cn ct t gnt Lwis cids nd, in tun, duc th cntst ti bf psu t th dsid ditin. Th unifmit f th thml ditin in th bing pcss nd th gdul mping f th tmptu is hwv vitl in th btmnt f th inscpbl sufc ffcts. Th smpls thn psd t th dsignd intfnc pttn twic, nc t n 0 dgs, nd nth t th ngl f chic but th ptic s. Bf psu, th intfnc stup is lignd fm dift cusd b vibtin cupling nd minut thml vitins. T dmnstt, tw ngls w pimil chsn f th scnd psu, 90⁰ nd 60⁰. Th thgnl psu bing fundmntl in th dsign hibits ducd cstl smmt nd in n ttmpt t cct f this, nth psu is dmnsttd. Th 60⁰ psu ttin ccts f sm f th dstd smmt sulting fm lngtin in th stcing dictin. Th dutin f psu is pimntll dtmind nd is fund t b ptiml t 0.9s f bth ttins. Th psd smpls w thn bd gin f 0 minuts t 65⁰C nd f 5 minuts t 95⁰C in d t csslin th gins tining significnt munts f Lwis cids. Aft cling t m tmptu, th smpls w plcd in th dvlping slutin f 5 minuts nd wshd ff with isppl lchl nd llwd t d in i whil mintining lw intnsit d light nvinmnt f th nti pcss. 55

66 4.0 XPRIMNTAL RSULTS 4. STANDARD VALUATION Thugh pimnt, pcdu ws dvlpd tht dptd t th til nd inhnt in th pcss. Th munt f SU-8 fmultd ws such tht it pvidd nugh smpls tht th ittins ndd t pduc cnvgnc t lcl ptimum culd b chivd in windw f but 0 ds. This ws ughl stimtd t b but 00 smpls ut f which t lst n smpl tht shwd pgss ws btind. Aft 0 ds, nw slutin ws ppd s th dditinl unctint cquid b dgdtin f th slutin t n ld isting suplus ws undsibl. Initil pimnts w cnductd withut th dditin f n cid scvng; instd, th munt f pht initit ws vid with ptil succss. Aft svl ittins sm f which pvidd cllnt sufc cdings, nl fw hd pnings nd fm this fw nl fctins hd pnings bnd th fist l. Initill gl phs ms muntd n ttin stg ws utilid dding dditinl unctint t th psitining f th scnd intnsit distibutin nd hncfth incg th numb f tils ndd t cnvg t smwht ccptbl sults. Th smpl tht ssud cnfidnc s wll s inititd th sch f n ltntiv mthd f cntl v cid diffusin cm ft mn tils with full pnd ftus nl n l dp. 56

67 Figu 5. Sufc SM img f smpl hving 60 stuctu A fw ittins sultd in smpls with v unifm pnings but w cnfind pul t th sufc. Imging nlsis with th dpth in scutin shwd tht vn ft ds f dvlping, th mining unpnd stuctu ws nt ffctd. This suggstd tht ith th psu tim ndd t b shtnd th bcgund ng ndd m cful ttntin. Figu 6. Sidviw SM img lft shwing ptil pning, Sufc pticl micscp img ight shwing cllnt sufc stuctu 57

68 An ltntiv mthd f cntl v th bcgund ng ws ccmplishd b th dditin f n cid scvng, TA. Vius SU-8 slutins w md cntining t 40 ml% TA ch f th pdtmind vlum. Slutins tht cntind 0 t 40 ml% TA pvidd stuctus with full pnd ftus f which nl fw did nt cllps ptining t v sht psu nd dvlping tims. Th pcisin tht ths slutins dmndd in th psu nd dvlping stgs sultd in thinn thn pctd ftus suggsting tht th pht-cids w nutlid m thn dsid. Figu 7. SM img f 60 stuctu, lft cls mintin f th sufc, ight mintin f th sid A smpl cquid b ping cful ttntin t th psu nd dvlping nd futhm t ding ws btind nd full cllps ws pvntd. This stuctu hibits stuctul intgit n lg scl; hwv, t mining th dpth dimnsin, th ptil cllps tht t plc is ntbl. This is sn s th sidws swing f th img nd it mst lil t plc in th ding pcss s it is h tht th stsss td n th stuctu culmint fm th cpill fcs btwn th liquid nd i intfcs. 58

69 Figu 8. SM img shwing full pnd cstl with sufc viw ight nd sid viw lft Th mining slutins cntining t 0 w% TA, 0% lng with 0.9s psu tim ildd full pnd stuctus withut n ntbl cllps. Th SM imgs tht mgnifid t vl th stuctu cntin insts ptining t tht which is pdictd b simultin. Th simultin is gntd in Mtlb b mpping ut th vll intnsit distibutins nd pltting th nmlid isintnsit vlum f 0.5 nd bv. Th Mtlb pgm is fund in Appndi B. Th th-dimnsinl simultins thn ttd such tht th cstl s lin up with tht f th SM imgs tn. This stuctu psssss mdifid FCT smmt nd is th sult f 60 ttin btwn th fist nd scnd psus. 59

70 Figu 9. Sufc img shwing lng ng d tp lft, cls viw with pdictd stuctu inst tp ight, sid viw with pdictin inst bttm lft, tp viw f nth cstl pln bttm ight Th pidicit f th tmplt in th -dictins is stimtd b msumnt tls built int th scnning lctn micscp imging sstm. In this dictin, th lngths qul in mgnitud nd th vg f fu msud vlus ws µm, whs th stimtd vlu ws.05µm. Pfming th sm msumnts n th dpth dimnsin gv th msud pidicit C t b nwh fm.8-5.5µm s ppsd t th 5.µm stimt. This msumnt might nt b s ccut s it is difficult t sctin th ttin f th tmplt. Pi hdships w ncuntd with th ttin f th smpl hlding stg t th ct vticl psitin. 60

71 4. DIFFRACTION ANALYSIS Th smpls tht w imgd nd dmd t pssss cllnt sufc ftus, full pnd stuctu with ugh sufc, cmbintin f bth w thn subjctd t diffctin b Hlium Nn ls nd whit light suc. Althugh th implmnttin f th diffctin nlsis did nt vl n quntittiv sults, it is hwv fund t qulittivl ssss whth th stuctus pssssd glbl lcl stuctul unifmit. Ittins pi t th dditin f TA pvidd smpls tht hd v lg s f dfct f gins limitd t th sufc. Ths f cus hd v cln diffctin imgs whn subjctd t HN ls nd whit light suc. Figu 0. Diffctin img tn f 90 stuctu with HN ls lft nd whit light ight Ittins tht pvidd smpls with full pnd ftus in th-dimnsins gnll hd ugh sufc fm cllps s thin ptuding mmbns did nt hv n ltl suppt t withstnd th fcs f sufc tnsin. Th diffctin imgs fm ths smpls w pllutd with light spts s th ndm sufc scttd sm f th s ff s. 6

72 Figu. Diffctin img tn f 60 stuctu with HN ls lft nd whit light ight A smpl pcssd with th slutin mitu cntining 0 ml% TA with n psu tim f 0.9s pvd t hv th gtst chnc f pviding smpls tht hd bth full pnd ftus thughut th stuctu lng with cln sufc ftus if th dvlping nd ding w cntlld v cfull. Cful dvlping ws dn thugh cntinuus immsin int th tw pti dishs plcd sid b sid, n cntining th SU-8 dvlp, th th th isppl lchl. This llwd th mvl f wshd w cmpnnts tht culd ptntill dh t th sufc f th smpl. Figu. Diffctin img tn f 45 stuctu with HN ls lft nd whit light ight 6