Faculty STI Section of Microtechnology Institute of Applied Optics Mars 2002 to June 2002

Transcription

1 IDNÖSSISC TCNISC OCSCUL LAUSANN POLITCNICO FDRAL LOSANNA SWISS FDRAL INSTITUT OF TCNOLOY LAUSANN Faulty STI Sto of Mothology Isttut of Appld Opts Mas 00 to Ju 00 Computato of ustom mad photo ystals Smst pojt Mal Lutgg Swss Fdal Isttut of Thology Lausa (PFL) Swtzlad D. Ivo Ut Swss Fdal Isttut of Thology Lausa (PFL) Swtzlad Pof. Olv Mat Swss Fdal Isttut of Thology Züh (TZ) Swtzlad Pof. Pat offma Swss Fdal Isttut of Thology Lausa (PFL) Swtzlad

2 IDNÖSSISC TCNISC OCSCUL LAUSANN POLITCNICO FDRAL LOSANNA SWISS FDRAL INSTITUT OF TCNOLOY LAUSANN Faulty STI Sto of Mothology Isttut of Appld Opts Mas 00 to Ju 00 Abstat Th pst pap otats o th thotal udstadg ad th umal omputato of photo ystals. It povds a toduto to th bas thoy shows som omputato sults ad th tptato by mas of a fw sampl ystals ad poposs som masumt mthods to valuat th pdtd bhavou. Th tstd us fds som addtoal softwa fomato a oupl of omputato sults ad som mags of a al mad photo ystal th ax. I ths wo th MIT Photo-Bads softwa paag was usd fo all of th umal omputatos. Th softwa was sussfully stalld o a psoal wostato at hom ad o a sv at th sttut. Its apablts ad ts futoalty w td out. Th pogam was judgd to wo labl. It ylds th mo auat sults th mo tm was spt fo th omputato ad t fds patdly th sam sults fo th sam tas v o dfft mahs ad pogam vsos. Th th sults w ad to MatLab fo powful posttatmt ad gaphal pstato. Fo ths aso som slf-pogammd MatLab futos a also appdd to th ax. A total of th photo ystal sampls w dsgd by umal omputato ad layout. ah sampl ovs a patula futo: a) a smpl bad gap flt b) a hgh-q avty ad ) a tgatd wav-gud. It was show that a hgh dlt otast s dsv to dsg lag bad gaps that a qut obust agast fabato os. Bfo spdg mo tm fo smulato ths wo may b otud by fabato ad masu of th dsgd photo ystals. Lausa May 3 00 Th autho: Mal Lutgg Aowldgmts fo asssta to: D. Ivo Ut Kay assl Pof. Mat Pof. offma Pof. Lass

3 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal Tabl of otts Pojt bf 4. Itoduto 5. Thoy 6. Bas poblm 6. Dspto of th ystal stutu 7.3 Radato mods gmods of th photo ystal 7.4 xpaso of th gvalu quatos 8.5 Poof of Bloh s thom 9.6 Salg laws 0.7 Numal omputato.8 Ral vsus poal latt.9 Polasato 3.0 Dsty of stats 3 3. Photo bad stutus 5 3. Bad dagams 5 3. Itptato Squa latt Tagula latt 7 4. xampls of omputd photo ystals 9 4. Squa latt wth smpl ut ll 9 4. Tagula latt wth smpl ut ll Photo ystal sampls Absopto & lmtd sz 5. Layout of photo ystal sampls 3 5. Fabato 3 6. Poposd masumt mthods 3 6. Tasmsso mod da fld mosopy 4 6. Sufa sattg of a N las ba lght Ba sattg ad tf of a N las ba lght 4 7. Summay 5 8. Rfs 5

4 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal Pojt bf Th thologal adva th matusato of ltos ass fo w matals always mo pfomat ad thfo pushs th sah ths doma. By xplotg th fomal aalogy btw lto ad photo sahs sudd dsgg ad podug matals wth astoshg optal popts: th so-alld photo ystals. Idd thy ould fom th bas of futu zo thshold lass tgatd optal fbs ad low loss mos. Ths may allow ovomg th d fo sgal amplfato at th d of optal fbs gat adva tlommuatos though possbl applato xtd o th t doma of optoltos. By umal smulato ths pojt ams to omput ad modl th optal popts of bdmsoal photo ystals gow o a pla odutv sufa by lto-dud dposto of a gas puso. Ths ystals may b dsgd as fquy flts (bad gap) wav-guds ad so o. Thy a b dpostd wth th sttut s sag lto mosop. So th ustom a fly hoos th ystal dsg povdd h dos ot as fo fatu szs blow 00m. Tass: Smulat th bhavou of som photo ystals modl th ffts ad pass th sults to M Ada Bahma fo th fabato of som tst sampls. Fally t should b possbl to vfy th modl by masug th podud sampls. Thoy: 70% Paxs: 30% Studt: Suppot: Mal Lutgg mal.lutgg@pfl.h Ass. Ivo Ut vo.ut@pfl.h BM 4.09 Pof. Olv Mat olv.mat@fh..thz.h TZ 6 Pof. Pat offma pat.hoffma@pfl.h BM 4.7 Dlays: Rpot a had out utl 7 00 o Ju 00 to Pof. offma Pof. Mat ad M Ut Pstato 5 pstato followd by 5 dsusso bg at 4 30 o 6 Ju 00 oom BM 5.0 at ampus 4

5 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal. Itoduto Th phaps fst photo ystals a ommoly told Bagg fltos. Although th thotal galsato to multdmsoal stutus was do th 960 s... th solutos ould ot b xtatd aalytally a qut xat ma. Thus t dd th appaa of powful psoal omputs th 990 s to ma umal alulato fasbl at asoabl osts ad omputato tm. Nowadays photo ystals a asly dsgd though umal smulato but th alsato ad tst bam th tal stp. At som xptos th xpmtal wo was mad th fa fad o v mowav ag at wavlgths abov 0µm. I otast th ystals mad dug ths pojt a ssd to wo th vsbl ag patula at th N las wavlgth of 633m. I osqu th fatu sz s of th sam sal thus about 00m to 500m. Stppg dow th wavlgth to lss tha /0 th of th usual valus xpts aothology fo fabato hgh-soluto mosopy/tfomty dug tsts ad ot at last oth optal matals taspat fo th vsbl lght. O o had ths omplats wo but o th oth had at last th tsty dstbuto th photo ystals should b dtly vsbl by mosopy. What a photo ystals? ally popl ow about th sold-stat physs xplag fo xampl th dffs btw lt solatos (ams damod...) smodutos (S aas...) ad odutos (mtals). Most of th ltal phoma ddu fom th gula ad pod aagmt of atoms ojuto wth th atom s hmal popts. Ths dfs th lto bad stutu of th mtod matals all ystall ad thus th lt popts. Aalogous photo ystals a mad of sval matals a gula ad pod aagmt. Whas th ltos a ystal tat wth th Coulomb pottal of th atoms th photos a photo ystal fl th fatv dxs of th matals. Ths lads to a fomally aalogu dspto of th optal bhavou tms of a photo bad stutu. Not that th optal popts a otollabl by th patula dsg of a photo ystal xatly l a smoduto s ltal popts a otolld by th atom stohast ad th spatal aagmt. Pojt goals Udstadg th opts patula th bad stutu ad th assoatd aalytal ad umal modls. Numal omputato of D photo ystals. Co-opato wth M Ada Bahma fo th fabato of som patula sampls ad th masumt. Compaso of th masus ad th omputd pdtos f possbl a quattatv ma. Famwo Itato of a w sah doma at th sttut. Implatato ad tst of som bas softwa tools. Povso of a "photo ystals shothad". Applatos Low loss mos as galsd Bagg fltos xtdg th wog ag fo th th d agl o th wavlgth. Ths may b ahvd though b- o tdmsoal photo ystals. gh Q las avts fo zo thshold lass. Wav-guds fo tgatd opts allowg vy aow uvatus. gh Q wavlgth flts. Could b usd tlommuatos to splt up a wavlgth-multplxd sgal to ts hals. Itato of soat stutus wth moluls. May b usful fo th dtto ad dtfato of spuous hmals o bo moluls pst a lqud. 5

6 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal. Thoy Th tato of a adato fld wth matt s th fudamtal phoma xplotd photo ystals. It allows th dsg of th ystal to otol ts optal popts a xhaustv ag. I fat t too som tm utl sahs otd that th flow of photos s otollabl a ma smla to th flow of ltos smodutos. Baus of th stog aalogy btw smodutos ad photo ystals th ad s xptd to hav just a lmtay owldg of ltomagtsm quatum mhas sold-stat physs ad omplx aalyss. Ths hapt gvs a toduto th thoy usd fo th pst wo. Fa fom bg omplt th tstd ad fds dtald fomato th ltatu.. Bas poblm I a matal wth o f lt hags ad o lt ut th Maxwll quatos fo ltomagt flds lad to th followg lmholtz wav quatos. Patulaly f spa: ( ( t )) ( t ) t wh s th vloty of lght f spa () ε µ 0 ( ( t )) ( t ) t () If lght tavls matt ts lt dsplamt s multpld by th latv dlt ostat ε ( ). At optal fqus magt matals a a. So ths wo xlusvly tats dlt matals wth a latv magt ostat assumd to b µ ( ). th lmholtz quatos tasfom to: ( t ) t ( ) ε t ( t ) ( t ) ( ) ε t Sg ths quatos by t ( t ) t ( ) ad ( t ) ( ) (5) lads to th agula fquy ad th ospodg spatal flds ( ) ad ( ) []. Th th dspso lato d vg d (wh v g stads fo th goup sptvly gy vloty) a b dvd ad th dsty of stats appoxmatd. Not that th agula fquy s a global ostat. Ths mas that oly last sattg s osdd []. Th agula fquy s fat a gvalu of th wav quatos ad ts ospodg flds a th latd gfutos. ths gfutos should satsfy th followg gvalu quatos: ( ( )) ( ) (6) ε( ) ( ) ( ) (7) ε( ) Dsgg a photo ystal s quvalt to layout a matal wth a pod ε ( ) suh that th solutos of (3) ad (4) follow th dsg spfatos. I th as of smpl futos l a susodal ε ( ) t s poss- 0 (3) (4) Th flds a otd as phasos thus a lt o magt fld magtud ad a phas shft a assgd to ah posto. I fat last sattg o fluos do afft th agula fquy but a ot dald wth ths pap. 6

7 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal bl to pfom a aalytal study. Fo a gal ε ( ) oly a tatv umal dsg poofd fasbl. fo a gv pod ε ( ) umal alulato xtats solutos a spfd fquy ag. I osqu ε ( ) s adaptd ad th alulato s lauhd utl a sutabl bhavou has b ahvd.. Dspto of th ystal stutu v a pod matal t a b fully qualfd by th dfto of ts bas patt ad ts podty. Ths dspto wll show up vy gaful baus t automatally mmss th umb of paamts fo umal alulato. Imag shows a bdmsoal ystal. Th ystal stutu s pstd by a st of bas vtos ad a ospodg patt. Th patt s smply opd at ah loato ahabl by a tg la ombato of th bas vtos. Fo ay tg ad l T wth T a lb (8) b d pots to a taslato gd pot a wh to opy th patt to. Not that th bas vtos ad th patt a ot uqu. Th patt s alld ut ll whas th taslato gd s ow as th ystal latt. Thy a pmtv f fo vy pot ad fom wh th ystal has th sam stutu (8) s aomplshd. Th ut ll flls up th t ystal volum by ptto o th latt. Imag : xampl of a D pod stutu. Th bas patt s boud o th taslato gd spad by th vto pas a ad b sptvly ad d. It s patd at ah gd pot. Not that th pmtv latt pots ad th pmtv ut ll volum a vaat to ay st of latt vtos o hooss. Imag shows two of may possbl hos: ) th vto pa ( a / b ) ad ts ut ll attahd o th gd pots at ts low lft o ad ) th vto pa ( / d ) wth a ut ll attahd somwh ts go. Imag : Sampl D ystal Imag 3: Sampl D ystal Imag 4: Sampl 3D ystal.3 Radato mods gmods of th photo ystal Th gvalu quatos (6) ad (7) showd that ay popagatg ltomagt wav s a gmod th photo ystal. Oth mods aot tasvs log dstas dd thy a vast mods [3]. 3 Th matal s osdd to b f of absopto. Also dffuso s ot pst ths mas that oly th vaato of 7

8 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal As th ystal has a pod futo ε th Bloh s thom [4] a b appld to th gvalu quatos. Thfo ( ) ad ( ) a haatsd by a wav vto th fst Bllou zo ad a bad dx : ( ) ( ) u ( ) wh u u T ( ) (9) ( ) ( ) v ( ) wh v v T ( ) (0) fo ay latt taslatot Baus of th podty of th photo ystal ad th sam podty of th solutos t suffs to ow th flds o pmtv ut ll. Th global soluto s th gv by smpl plato ov th ystal latt. Baus of th flds podty ad otuty thy hav to b qual o oppost boudas of th ut ll. Th of a wav tavllg though th photo ystal. Itutvly t dos ot astosh that th wav s fld podus th ystal pod suh that t bhavs th sam ma ah ut ll t outs. I patula ths holds fo o-absobg ystals. But v f th wav s absobd t attmpts to podu th ystal pod whl loosg tsty..4 xpaso of th gvalu quatos Baus ( t ε ) s pod ad ay soluto ( t ) ( ) has th sam spatal pod th gvalu quato (6) a b xpssd tms of Fou ss. Fst gals quato (8) to th dmsos: 3 T l a wh a a th latt vtos ad l Z () π I o dmso th Fou ss would b xpssd tms of th bas spatal fquy b ad a ts hamos lb. I th dmsos ths s wtt as [5] : 3 l b wh b a th poal latt vtos ad π f j suh that a b j πδ j 0 ls Th dlt futo ε ( ) ad quato (9) a ow xpadd Fou ss: κ ε ( ) wh ( ) l Z () (3) κ a th Fou offts of ε (4) wth th Fou offts Baus th sults a sahd tms of pla wavs [s quato (9)] ths mthod s alld th plawav xpaso mthod. Of ous solutos a b foud by oth bas futos suh as sphal wavs. Idd th sphal wav xpaso mthod has povd ts advatags fo th umal alulato (5) 4 5 th latv lt ostat satts th wav. Th thom holds fo photo ystals too. It ogats fom th dspto of th lto bhavou th pod lt pottal du to th gula aagmt of atoms ystall solds. Bloh statd that th gfutos of th Shödg quato fo a pod pottal a b xpssd as poduts of pla wavs wth ospodg futos u ( ). Ths Bloh futos u ( ) dpd o th wav vto ad a pod th ystal latt so that u T ( ) u fo ay latt taslatot. S ".8 Ral vsus poal latt" o pag. 8

9 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal 9 of photo ystals wth yldal o sphal shaps ε. To omplt th xpaso quato (6) s fally wtt as: ε κ κ κ κ κ (6) Th lft had sd of quato (6) s omputd up to a sufftly lag umb N of ad a gvalu poblm s solvd to xtat th gvalu. So th dspso lato of th photo ystal s obtad by omputato of tstg fquy bads at som pots..5 Poof of Bloh s thom Fst th gvalu quato (6) s wtt as: ε (7) To pass to th Fou doma th dlt futo s xpadd a Fou ss ad th lt fld s gfuto as a Fou tgal: d A d A d A d A d A d A d A ε ε ε ε ε 0 d A A ε (8)

10 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal Ths has to hold fo all posto vtos so th tgad ds to vash: ( A ) ε A 0 ( ) quato (9) mpls that all Fou ompots A( ) ot latd to th poal latt vtos a zo. Ths mas that th lt fld a b xpssd a Fou ss stad of a Fou tgal. Thfo: ( ) A A u (0) Th lt fld s pod wth th sam pod as th dlt futo. I gal th umb of Fou ompots s ot lmtd so that quato (9) xpads to a ft st of gvalu quatos. th gvalus ad gfutos a dxd by a subspt. quato (9) has b showd. Of ous quato (0) a b povd a smla ma..6 Salg laws Th a som usful popts of photo ystals. t wll b show that a ufom salg of th spatal pod th tm o th dlt futo sults a ufom salg of th ystal s gfqus. Fst gv a ufom sal of th dlt futo: ε ( ) mε ( ) wh m R > 0 () Istd to quato (7) lads to: ( ) ε ( ) ( ) () By ompaso wth quato (7) th sults a b dvd mmdatly: ad ( ) ( ) (3) m (9) Sod gv a ufom sal of th spatal pod ad th tm. If th sal fato a s th ystal s latt sz th w vaabls L ad t a dmsolss: L ad t t wh a R > 0 [m] (4) a a Th lmholtz quato (3) s tasfomd to: ( t ) ( t ) wh dots th dfftato wth spt to ε t (5) Ths quato mpls that ts solutos a b laly tasposd to aoth ystal dffg oly lgth but ot stutu. a spf stutu ds to b omputd o ad holds th solutos fo ay sald stutu. By ovto th omputato s do fo th omalsd -vtos ad sults th omalsd gfqus. Thy a th sald to th osdd ystal by: 0

11 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal π π ad wh th fato π has b std by ovto [6] (6) a a.7 Numal omputato So fa th aalytal omputato of th dspso lato a photo ystal has b showd. Fo umal omputato th quatos d to b dstzsd. quato (6) suggsts a lgat soluto. It lats th Fou offts κ ( ) of th vs dlt futo ε ( ) (4) to th Fou offts of th gfuto ( ) (5). Fst th volum of th ystal s ut ll s dvdd to N volum lmts. Fo ah lmt ε ( ) s dtmd [7]. Th th dst Fou tasfom ylds th N offts κ ( ). Thus quato (6) xpads to N la quatos at N offts ah. A omplt soluto would ma to dagoalz a N matx a tm osumg tas v fo modat N ad just mpossbl fo bg N! Fotuatly most ass a patal soluto say M gvalus ad gfutos wth M << N s sufft fo th aalyss o th dsg of a photo ystal. A subst of M gvalus a b omputd muh fast usg a ovgg tatv gsolv. Not that th soluto of quato (9) by us of quato (6) luds spuous zo mods that a ot pst fo quato (0). Idd quato (0) matas tasvsalty [8] thus lmatg th zo mods. Advatagously alulato s do fo th magt fld. th lt fld a b dvd by popotoalty..8 Ral vsus poal latt To show th xpaso of th gvalu quatos th poal latt was mtod ad ts latt vtos w gv by quato (). Ths sto dtals th mag of th poal latt ad shows how t s ld to th ystal latt. Rmd th ommoly ow Bagg gds. Idd thy a D photo ystals. Th mag 5 shows two xampls wth qual pmtv latts but dfft pmtv ut lls. a ah ut ll flts pat of th dt lght du to th modfato of th dlt ostat. Th total flto s patulaly hgh at all wavlgths that podu ostutv tf btw th fltd pats. O th oth had th total tasmsso s patulaly hgh at all wavlgths that gv dstutv tf btw th fltd pats. It may astosh that both Bagg gds flt lght maly at th sam wavlgths. Idd th ut ll patt dtms sstally th flto ffy fo wavlgths aoud th xat valus wh thy thmslvs a dtmd ppally by th latt sz a ad th volumt ma ε ( ). Imag 5: a a Two Bagg gds th o of a optal fb. I ths xampl both hav th sam spatal pod a. both flt th d lght ad tasmt th blu o. Th pot gd blow tas th spatal pod but abstats fom th patt ad a omalsd to uty. Th fquy s gv by f a ad th wavlgth th ystal by λ. a Ths s a tal tas! Th soluto s auay ad th gsolv s ovg spd a stogly modfd by th ma of how th vs dlt futo s attbutd to th volum lmts. S sto.3 "Blo-tatv fquy-doma mthods fo Maxwll's quatos a plawav bass" fo dtald fo. Th magt fld s always ppdula to th vto. S th poof at sto. "Optal Popts of Photo Cystals".

12 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal Baus of ts domat ol th latt wll b adjod wth ts poal pstato th -spa. I th as of th la latt a ts poal latt s also la wth th latt vto: π a a b π so that a b π (7) a a a a Th ystal latt s a latt al spa. Its latt vtos a uts of lgth [m]. But th poal latt s a latt th -spa wth vs uts [m - ]. Fo xampl mag a pla wav popagatg th latt dto: ( t ) ( t ) 0 wh a ad b (8) ( z t ) 0 z π t λ Th sala podut th xpotal s a podut of a posto vto th al latt wth a wav vto th poal latt. If th wav fld s ow oly at th latt pots a wav wth wavlgth λ aot b dstgushd fom ay oth wth wavlgth: 0 aλ0 a λ0 λ wh Z (30) Istd to quato (9) t a b s that th xpotal tas th sam valu o th latt pots at z ma wth m Z. I fat ths s just a mafstato of Shao s samplg thom. I -spa t a b wtt as: b b 0 b wh Z ad 0 (3) Th ag of 0 s th so-alld fst Bllou zo [9]. I th poal latt th fst Bllou zo B 0 s othg ls tha th ut ll that ylds: B0 hagof B0 db m (9) wh B 0 s th ostat volum (3) ad B0 th vaabl shap of th poal ut ll Th opt of th poal latt a b appld to ay 3D latt as t was alady mtod th quatos () ad (). Th sala podut of a latt taslato wth a poal latt taslato ylds: T l a m jb j Istg quato (3) lads to: T π l m π tg j j la m jb j (33) baus l m Z (34) Ths mpls that pla wavs wth ' s dffg by pst th sam flds o all latt pots T. Idd quato (3) a b galsd to: 0 wh 0 B0 (35) Not that th dfft wavs hav dfft gs. v f thy sm to b dtal o th latt 9 Ay wav wth b b sampld at th latt pots a b tly ostutd by tpolato.

13 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal pots th flds th to of th ut ll a wll dfft. Idd th opt of th poal latt ad th Bllou zos wll b vy hady to pst th dspso lato a ompat ma [0]. v th latt vtos th poal latt vtos dv fom quato (3). Fo xampl th vto b stays ppdula to th vtos a ad a 3. t has to b paalll to a a3. Its pojto o a has to b π thus a C( a a3 ) π. Solvg fo C ylds: a a3 a a3 b π π a a a3 V0 a3 a a3 a b π π wh V 0 s th volum of th ut ll (36) a a3 a V0 a a a a b 3 π π a3 a a V0 Oft th ostat π s omttd. So b aful wh tptg th poal latt! Fo D latts a thd latt vto 3 a a s todud to apply y a z y 3 th quatos (36). Th thd poal latt vto b 3 s th paalll to a a b 3 although maglss. Imag 6 shows a smpl D tagula latt. Not that th fst Boull zo B 0 s th sufa aoud th 0 a x x V 0 b B latt og. It xtds just to th fst of th ppdula bstos that a daw btw th latt og ad ts ghboug latt pots. Ths s th gomtal quvalt of odto (3). Imag 6: Squa latt wth latt sz a ad ts poal latt also squa wth latt sz b. O of th ut lls ad th fst Boull zo a gyd..9 Polasato I D photo ystals th ystal stutu s sttly quvalt fo both th lt ad magt fld. th solutos do ot dpd o polasato. I D photo ystals th dlt ostat vas a pla. Now th ystal stutu dffs fo wavs wth dfft polasato. Nvthlss all solutos a b obtad fom a supposto of two patula la polasatos: a tasvs lt (T lt fld pla) ad a tasvs magt (TM magt fld pla) wav. Idd th lt fld of th TM wav s always paalll to th stutu vaatos. t has to b otuous aoss th matal boudas. O th oth had th lt fld of th T wav s oly patally paalll to th stutu vaatos. ts ppdula pat d ot b otuous aoss boudas. I 3D photo ystals sults dpd o polasato. th polasato s osdd to b adom thus oly yldg solutos fo ay polasato..0 Dsty of stats Th dsty of stats a b adly omputd out of th total umb of mods up to a spfd quatum gy. I f spa th total umb of mods up to a gy of D 0 th volum V 0 s gv by: V 0 N( 0 ) d 3 π 3 V0 4 0 π V 3 3 ( π ) 3 3π < 0 wh two othogoal la polasatos (dpdt) w osdd (37) 0 S "3. Bad dagams" o pag 5. 3

14 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal Th dsty of stats s th: N V0 0 D( 0 ) 3 π 0 volum V 0 (3D) (38) S00 π o sufa S 0 (D) (39) L0 π o dsta L 0 (D) (40) I th gal as of photo ystals ostat quatum gy dos ot dsb a ula/sphal sufa th poal spa. Thus th total umb of mods ad th dsty of stats a mo dffult to omput. Nvthlss th dsty of stats a b appoxmatd f th spd of lght s plad by th ma goup vloty v g fo wavs of th gy D 0. 4

15 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal 3. Photo bad stutus Ths hapt dals wth th pstato of th dspso lato by so alld bad dagams. It should hlp to udstad how to ad out th haatsts of a photo ystal fo xampl bad gaps goup vloty o a appoxmato of th dsty of stats. 3. Bad dagams Rmd th dspso lato d d f spa. It ould b pstd by a staght l a --dagam l gaph. Now todu a pod hag of th dlt ostat th x- dto. Th modfd dspso lato d vg d ould b pstd th sam dagam. Ths wos f fo D poblms. But th as of D o 3D ystals how to pst th dspso lato fo dfft dtos th sam gaph? 0 Sto.4 showd that th dspso lato 0 π a splts up fquy sub-bads ( ). Ad sto aph :.8 todud th fst Boull zo assgd to th poal latt. th spatal fqus π a th sod ad 3 3π a th thd Boull zo ospod to π a th fst Boull zo! Ths s quvalt to fold gaph at th Boull zo boudas. Th sult s th so-alld bad dagam showd gaph. By ovto fo tg th spatal fqus π a a told th gamma pot Γ ad th X pot stads fo ( 0 ) π a.5. Th dashd ls dat th bad stutu of a homogous matal whas th th ls show th fst fou bads of a sampl Bagg flto. Cauto: B aful wh ad out -valus baus th -axs s ot ssaly sald ufomly! I ths pap th salg s always th gomtal path lgth appag th poal latt. 3. Itptato slop 3 slop v g π a 3π a Dspso lato f spa ad fo a D photo ystal (Bagg flto). 4 0 Γ aph : 3 4 X Bad dagam Loo at th xampls gaph ad mag 8. A bad dagam dtly shows a pat of th dspso la- o th bouday of th dubl Bllou to of a photo ystal. Imag 8 shows th lato of ( ) zo. If a pla wav wth fquy s dt o th ystal o of th pstd dtos th bad dagam foms about ts spatal wav vto. If th fquy falls a mpty ag ts dto a so alld patal bad gap th ystal dos ot apt a popagatg wav of that fquy ad dto. Idd th ystal would add a magay otbuto to thus fog th wav to b vast. Fst osd a popagatg wav. Th slop d d at th wav s ( / ( ) ) pot gvs th wav s goup ad gy vloty v g th ystal []. Whas th goup vloty s wll dfd th phas vloty s ot baus of th dffulty to follow th popagato of a spfd wav fot (ostat phas). As a xampl loo at th fst bad mag 8. Fo small aoud Γ th goup vloty s aly ostat all dtos. But as appoahs M o X th wav sgfatly slows dow. Supsgly t stops popagatg S sto.5 "Optal Popts of Photo Cystals" fo dtals o th lato of th phas th goup ad th gy vloty a photo ystal. 5

16 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal at th M ad X pots []. Suh "zo vloty" pots a alld bad dgs. Th lght thy aptu s sad to b foz. Sod osd a vast wav. I th ystal ths wav aot tavl though log dstas. t wll b fltd ath tha tasmttd. Th fft s vy smla to th total tal flto dlt wav-guds [3]. Fquy ags wh patula hgh flto ous a alld bad gaps. If th gap s pst fo ay dto t s a full bad gap (l th lght d zos mag 8) othws a patal o (lght blu zo). Bad gaps a always lmtd by bad dgs. Bsds aalyss shows that th wav ptats th dp to th ystal th mo ts fquy appoahs a bad dg..0 ow a w fgu out th apaty of th ystal to tasmt sptvly flt lght? Loo at gaph 3. Ths s just aoth pstato of th dspso lato mag 8. Th latoshp of tmpoal ad spatal fquy has b doppd favou of a summay out of aptd mods so alld stats. Th full bad gaps a of ous th zos wth o stats at all. O th oth had th xtmly flat bads at f 0. 9 show up a asd stat out. Th stat out s appoxmatly popotoal to th dsty of stats ad th ystals apaty of tasmttg lght wh th dspso lato shows at whh vloty tasmsso ous. f [/a] stats [outs] aph 3: Mods of th xampl mag Squa latt Ths latt s hghly symmt baus t s vaat to: - otatos of tg multpls of 90 aoud th og - fltos at th xy 0±0.5a ad x ±y axs - pot flto at th og Baus of symmty th fst Boull zo a b dud to a v small zo. If th dspso lato has b omputd fo th dubl Boull zo B th t a b xpadd by symmty o th t fst Boull zo. V 0 Imag 7: a y a x Squa latt wth latt sz a ad ts poal latt also squa wth latt sz b. O of th ut lls ad th fst Boull zo a gyd. b y B 0 b x 3 I paxs quatum mhas hbts xat owldg of ad. th wav s always popagatg v a pft ystal. If som absopto s pst t wll advatagously wa th bads dfto (sma thm out). Must ot b ofusd! Total tal flto ous at th bouday fom a 'ds' to a 'lght' dlt matal. I otast th ma dlt ostat of a photo ystal s gally hgh tha th o of th suoudg optal mdum. 6

17 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal Fo xampl th squa latt mad of yldal ods ps th latt symmty. th -pots X ad Y a quvalt ad d ot to b omputd spaatly. Spal pots Th gamma pot Γ : Γ 0 dots always th og of th poal latt. b Χ : Χ ± b Υ : Υ ± Μ : ± Μ 3.4 Tagula latt Χ Ths latt s also hghly symmt baus t s vaat to: - otatos of tg multpls of 60 aoud th og - fltos at th xy 0 axs ad th pla otatd at ±30 aoud th og - pot flto at th og Baus of symmty th fst Boull zo a b dud to a vy small zo. If th dspso lato has b omputd fo th dubl Boull zo B th t a b xpadd by symmty o th t fst Boull zo. Υ b Υ Γ Imag 8: V 0 Imag 9: B Μ Χ B 0 b f [/a] M X M a) Fst Boull zo wth th dubl Boull zo B dad b) TM bad stutu of a photo ystal wth a yldal od of ft hght o vy latt pot [4]. Th ods hav a adus 0. 3 a ad a dlt ostat ε 9. Thy a st a wth ε. Th fst two omplt bad gaps a mad lght d a patal bad gap lght blu. Not th patulaly flat bads at f y a a x b Tagula (hxagoal) latt wth latt sz a ad ts poal latt also tagula wth latt sz b. O of th ut lls ad th fst Boull zo a gyd. y b B 0 x 4 S mag 7. 7

18 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal Fo xampl th tagula latt mad of yldal ods ps th latt symmty. Spal pots Th gamma pot Γ : Γ 0 dots always th og of th poal latt. b b b b M : M ± ± ± b b b b K : K ± ± 3 3 b b ± 3 Γ B b Κ Μ f [/a] b 0.0 K M K Imag 0: a) Fst Boull zo wth th dubl Boull zo B dad b) TM bad stutu of a photo ystal wth a yldal od of ft hght o vy latt pot [5]. Th ods hav a adus 0. 3 a ad a dlt ostat ε 9. Thy a st a wth ε. Th fst th omplt bad gaps a mad lght d a patal bad gap lght blu. 5 S mag 9. 8

19 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal 4. xampls of omputd photo ystals Ths sto psts som alulato sults ad th tptato. Rmd th goal bhd ths omputatos: pdt th bhavou of som sampl D photo ystals dsgd fo us at th N las wavlgth. Th sampl stutus had b sttd to squa ad tagula latts fo omputatoal as [6] as wll as to stay ompaabl wth oth wos. A oupl of gaphs a b osultd th ax. 4. Squa latt wth smpl ut ll Th followg sults w omputd fo squa latts wth sz a ad a sgl yldal od ah ut ll. Th od s paalll to a a has a adus ad s mad of a dlt o-absobg matal wth dlt ostat ε ε ε 0. Th ut ll s flld up wth a. aph 4 shows th bad gaps futo of th a ato fo a hgh ε- otast of. Not th ps of sval tasvs magt (TM) bad gap gos whas th tasvs lt (T) bad gaps a aly abst. I th 0 th lowst bads th s o full polasato dpdt bad gap. Th lowst TM bad gap go s spad by th st ad d tm bads. It has ts lagst latv ag fo 0. a. Rmd th mthod of omputg th photo bads: t has b showd that th ε futo s sampld ad that th fast Fou tasfom s ta. Th xtt of th lowst bad gap dpds maly o th fst Fou offts pstg th ma valu ad th ma fquy..4. Idd fo 0. a th ma fquy offt s a.0 ts maxmum to gv s to th fst bad gap s lagst xtt. Th ma valu dtms maly th ma bad gap 0.8 fquy. fo asg a dass ad th bad fqus do follow ths das. Just md that o a global vw th ma valu fatv dx. ε ε s gv by th ma Ta a loo at gaph 5 ow. A D photo ystal has always full T/TM bad gaps fo vy thss ato btw a ad hgh ε gos! Wll th T ad TM mods aot b dstgushd o dmso thy a both paalll to ay modfato of th dlt ostat. But otast to th D ystal th D ystal has a lot of lag bad gaps ovg total mo fqus tha th popagatg mods! Why s th pfoma of th D ystal so 'bad'? Idd a wav tavllg th D ystal fls a udtoal toato omg fom th patal fltos at th lay boudas. I otast a wav popagatg th D ystal fls a budl of wa fltos all dtos sattd at th boudas of th yldal ods. So th toato tas pla fo a t ag of pods a to a (x/y axs to th dagoal). Ths smooth toato aows th full bad gaps baus a wav a 'hoos' amog dfft popagato dtos to fd a popagatg mod. full D bad gaps a oly fomd at hgh ε-otasts fog th patal fltos! f [/a] V 0 a a Imag : Squa latt [a] aph 4: ap map fo ε. T bad gaps d ad TM bad gaps blu. f [/a] [a] aph 5: ap map fo ε ad a D photo ystal (la latt). T/TM bad gaps g. 6 Baus of th small ad smpl ut ll. 9

20 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal 4. Tagula latt wth smpl ut ll Th pvous sto showd that a squa latt dos ot off a lag T bad gap fo th matals that M Bahma ould us to buld up th photo ystal sampls. Thfo th latt stutu s hagd. Baus th hxagoal latt offs a v hgh symmty tha th squa o th full T bad gaps w xptd to b lag ad mo abudat. Th followg sults w omputd fo hxagoal latts wth sz a ad a sgl yldal od ah ut ll. Th od s paalll to a a a has a adus ad s mad of a dlt o-absobg matal wth dlt ostat ε ε ε 0. Th ut ll s flld up wth a. V 0 aph 6 shows th gap map. Not th gat umb of bad gaps ad th mo gula aagmt of th gap gos. I th 0 th lowst bads a th s a full polasato dpdt bad gap go at 0. 35a ad f 0. 8 a. As xptd som lag T bad gap gos a pst. A Imag : Tagula latt tstg T go s foud at 0. 6a ad f a. Not that th bads a pst at low fqus th tagula latt tha th squa latt. Ths s du to th mo ompat hxagoal aagmt of ods ladg to a asd ε h low fquy. Not also that th aly abs of lag T gaps s du to bouday odtos. Whas th TM polasato has ts lt fld paalll to th z-axs ad thus paalll to all matal boudas th T polasato has th xypla h a abtay otato to th matal boudas. Fom Maxwll s quatos t s ddud that th lt dsplamt D ompot ppdula to a matal bouday has to b otuous whas th paalll ompot s dsotuous gal. Th sults s that a D photo ystal th fquy ag fo popagatg TM mods s mo ostat tha th T ag. 4.3 Photo ystal sampls f [/a] [a] aph 6: ap map fo ε. T bad gaps d ad TM bad gaps blu. aph 6 showd th lagst T gap at 0. 8a fo a hxagoal stutu of yldal ods wth ε. Ths sto dtals ths patula ofguato. Bad gap sampl: wavlgth flt f45 f f [/a] 0.6 f [/a] K M K aph 7: T bad stutu fo th 8 th lowst photo bads 0.0 aph 8: Appoxmat dsty of T stats (abtay sal [s]) Th ystal psts a gap fom bad 4 to bad 5 wth a md-fquy f a ad a latv xtt 0

21 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal g %. Obsv th hgh dsty of stats fo bads wth a vy low goup vloty. Ths sps ould b patulaly tstg fo a low thshold las baus of th stog lto photo tato (absopto spotaous ad stmulatd msso) du to th log ps of th photos. Fo fabato of th bad gap sampl th N las wavlgth ds to math f 4 5. th latt ostat s a 0.9λ 580m ad th yldal ods hav a adus of 0.8a 05m. N Pot dft sampl: hgh Q avty f pot f pot f [/a] 0.6 f [/a] K aph 9: T bad stutu fo th 00 th lowst photo bads aph 0: Appoxmat dsty of T stats (abtay sal [s]) f [/a] V 0 a a K aph : T bad stutu aoud th two pot dft bads Imag 3: xagoal 5x5 supll wth o tal od (pot dft) Imag 3 shows th lagd ut ll of th tagula latt. Th ut ll patt s a xat opy of th bad gap sampl but t las a sgl od th t. By th ho of a 5x5 supll th pot dfts a xptd to b uoupld whas th omputatoal ffot mas fasbl [7]. Ufotuatly baus of th 5x5 supll th 8 mods of th bad gap sampl a foldd 5 tms ov ad gv s to 00 mods th sam fquy ag. It s possbl to otat oly o th dft mods ad th ghbous to du omputato tm. Futhmo f th dft mods a ompltly uoupld th goup vloty s xatly zo ths mas that th dft bad s ompltly flat. Thfo omputato of just o -pot suffs ppl. aph shows th fquy ag aoud both dft mods at f pot 0. 9 a. Idd th sdual goup vloty dats a (vy) wa ouplg btw th dft mods boud to ghboug ystal dfts. Th gy of th dft mod mas maly th to of ts ystal dft. Ths s du to th havy flto of ts wavlgth so that th photos aot sap. Th dft mods a b osdd as stadg wavs. Fo vy low absopto thy at avts wth a xtmly hgh qualty fato usful fo tlom lass o hoomts. 7 Th gd was st to a latvly small umb N8x8x. But omputato of 6 -pot too two days ayway.

22 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal L dft sampl: wav gud f [/a] 0.9 f [/a] K M K K' M' K' M aph : T bad stutu aoud th l dft mods. Th K' ad M' pots a -pots out of th l dft dto aph 3: Appoxmat dsty of T stats (abtay sal outs) Imag 4 shows th modfd 5x5 supll. Plas ot that th yldal ods hav a latv dlt ostat ε 0 ath tha. Ths typ of photo ystal a b usd as wav-gud fo tgatd optal dvs baus t suppots vy small uvatu ad to hag th wav-gud dto [8]. V 0 a a Imag 4: xagoal 5x5 supll wth o tal ods (l dft) 4.4 Absopto & lmtd sz If th dlt matal psts som absopto ths wll du th stgth of th toato du to th dud ma lftm of a photo. A smla fft a b xptd fo photo ystals wth lmtd sz fo xampl a t ut lls ah dmso stad of a fty umb. Th fft of wad toato wll b patula hgh fo photo ystals wth alady fw toato (low ε-otast). Th t wll sma out th photo bad fqus dug a gal das of bad gap wdth. Small gaps may dsappa ompltly. 0 0 Γ X Γ X aph 4: Bad dagam fo a D photo ystal. At th lft wthout ad at th ght wth havy absopto. 8 S paps o "

23 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal 5. Layout of photo ystal sampls M Ada Bahma usd a lto bam to dpost tps fom a vapoatd puso. Th dposto a b do o a odutv sufa. Thfo th glass substat was pvously ovd by a th ITO lay. Th otol systm dvg th lto bam pstd a adom latal shft. Ths dud gomt abatos ad mposd a upp lmt of dposto tm. th photo ystal had to b lmtd to lss tha a hudd tps. Th ystal s mad of aly yldal tps aagd gulaly o th ITO sufa. Th tp had s of ath llpsodal shap whas ts bottom s smla to a xpadg o. Th tps w gow at about λ 3λ.5µ m hght so that th vast fld s maly loatd th yldal ag. Baus of th sufaal loato th tps a b osdd as yldal ods of ft hght fo th vast wav. th photo ystal s osdd to b sstally bdmsoal. A N las bam tally fltd just blow th photo ystal ats th vast fld. Th wavlgth s λ 633m. Imag 5: lobal dsg of th photo ystal sampls It should b fasbl to masu th th sattg ( tasmsso o flto) o th phas shft dud by th total tal flto (shag tfomty wth th hlp of th ITO lay). 5. Fabato M Bahma usd TOS [9] as puso whh has a omplx fato dx ~ at th N las wavlgth. Fotuatly absopto s vy low baus th magay pat s lss tha a pt of th al pat of th fato dx. th omputato fo th sampl ystals was mad at ~ ε. 8 [0]. As mtod abov baus of adom mosop shft dug dposto th umb of ods p photo ystal had to b lmtd. Imag 6 shows a 9x5 hxagoal od aay wth at maxmum 4 tps. Th ystal would ov a sufa of about 3.6x6.0µm. By lavg out som mad tps th layout a b asly modfd to fom th pot ad l dft sampl stad of th smpl bad gap stutu. 6. Poposd masumt mthods Imag 6: Layout of th sampls. Pot dft wthout th wht od l dft wthout th lght gy ods too. It was pvwd to valuat th followg mthods xpmtally. Ths pat of th wo was doppd baus of th la of a wog photo ystal sampl. Nvthlss som patula mthods ad th xptd sults a pstd blow. 9 0 Ttathyl othoslat C 8 O O 4 S to fom SO x by lto dud vapou dposto. Th masud valu may hav som udfd basd o ths mas that th tu valu aot xd.8. 3

24 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal 6. Tasmsso mod da fld mosopy Ths da has ts aso a al wo [] wh a pot dft a photo ystal had b magd by flto da fld mosopy. It should b fasbl to wo tasmsso mod. Th fft of th photo ystal dft may b a olou shft at th dft ad a tsty hag sattd lght. Imag 7: Da fld mosopy to dtt pot dft. 6. Sufa sattg of a N las balght Basd o th xpmtal stup showd mag 5 th da was to ollt th sattd lght ov th photo ystal ad to dtm th postos wh patula hgh sattg ous. If th las bam ats a vast wav just o th lft half of th ystal sattg was xptd to b maly at th lft f th wav falls a bad gap of th ystal. O th oth had f th wav popagats th ystal sattg should mafst ath o th ght half. th th wav aot ally ptat to th ystal (sattg at th lft) o t popagats just to th ght bouday ad ds to lav th sufa baus th wav-gud stops (sattg at th ght). Idd f th ystal apts th wav t may just ot satt at all ths would b th as f th ps of th ystal dos ot modfy sgfatly th total tal flto. 6.3 Ba sattg ad tf of a N las balght Imag 8 shows th stup fo a shag tfomty masu. It s xptd that th apta of a mod by th photo ystal xpads ts tajtoy ad has th oos-äh shft. Th xpmtal wo should fgu out f th da s ot ad to whh xtt th phas shft foms about th haatsts of th ystal s bad stutu. Imag 8: Shag tfomty by th ITO lay S ax fo a opy of th atl "7x7 patt wth 600m dot-to-dot dsta Au-C pllas". 4

25 Computato of ustom mad photo ystals Lausa Ju 5 00 Lutgg Mal 7. Summay Th futoalty of th MPB paag was tstd ad foud qut powful. Th pogam was sussfully stalld o a psoal wostato at hom ad o a sv at shool. Som xampls of photo ystals w dsgd but ufotuatly ot yt podud ad masud. So th omputatoal o was valuatd by ovg of th omputato sults. Idd wth a plawav bass st of about 6 to 3 offts ah ystal dmso th alulato o was stmatd to ma blow 5%. I futu ths wo may b otud by: - fabato of th dsgd sampls - ompaso of omputd ad masud bhavou - ty out of som oth alulato softwa to quattatvly mbd absopto sz lmts masumt mthods ad so o. 8. Rfs - K. Saoda "Optal Popts of Photo Cystals": Spg-Vlag Bl/dlbg 00 - Chals Kttl "fühug d Fstöpphys": 4 th dto R. Oldbug Vlag Müh/W Stv. Johso Th MIT Photo-Bads Maual: vso.3 Massahustts Isttut of Thology vstd o Mah 00 - Stv. Johso ad J. D. Joaopoulos Blo-tatv fquy-doma mthods fo Maxwll s quatos a plawav bass: Massahustts Isttut of Thology vstd o Mah 00 - Photo bad gaps " vstd o Mah 00 5