Solitons in a system of three linearly coupled fiber gratings

Transcription

1 Solios i a syse of hree liearly copled fiber graigs rhr Gbeskys ad oris. Maloed Depare of Ierdiscipliary Sdies Facly of Egieerig Tel i iersiy Tel i Israel bsrac We irodce a odel of hree parallel-copled oliear waegidig cores eqipped wih ragg graigs (Gs) which for a eqilaeral riagle. The os proisig way o creae li-core G cofigraio is o se iered graigs wrie o ieral srfaces of relaiely broad holes ebedded i a phooic-crysalfiber ari. The objecie of he work is o iesigae solios ad heir sabiliy i his syse. New resls are also obaied for he earlier iesigaed dal-core syse. Failies of syeric ad aisyeric solios are fod aalyically eedig beyod he specral gap i boh he dal- ad ri-core syses. Moreoer hese failies persis i he case (srog coplig bewee he cores) whe here is o gap i he syse's liear specr. Three differe ypes of asyeric solios are fod (by eas of he ariaioal approach ad erical ehods) i he ri-core syse. They eis oly iside he specral gap b asyeric solios wih oaishig ails are fod oside he gap as well. Sabiliy of he solios is eplored by direc silaios ad for syeric solios i a ore rigoros way oo by copaio of eigeales for sall perrbaios. The syeric solios are sable p o pois a which wo ypes of asyeric solios bifrcae fro he. eyod he bifrcaio oe ype of he asyeric solios is sable ad he oher is o. The hey swap heir sabiliy. syeric solios of he hird ype are always sable. Whe he syeric solios are sable heir isabiliy is oscillaory ad i os cases i rasfors he io sable breahers. I boh he dal- ad ricore syses he sabiliy regio of he syeric solios eeds far beyod he gap persisig i he case whe he syse has o gap a all. The whole sabiliy regio of aisyeric solios (a ew ype of solios i he ri-core syse) is locaed oside he gap. Ths solios i li-core Gs ca be obsered eperieally i a ch broader freqecy bad ha i he sigle-core oe ad i a wider paraeer rage ha i cold be epeced. syeric delocalized solios fod oside he specral gap ca be sable oo.. INTRODCTION ragg graigs (Gs) are opical srcres i which he refracie ide is odlaed wih he spaial period eqal half he waelegh of ligh propagaig i i. The resoa odlaio gies rise o al coersio of coer-propagaig waes ia he ragg reflecio. The Gs are widely sed as opical filers sesors ad lipleers []. Crre applicaios eploi liear properies of he graig aely he presece of a gap i is specr which is idced by he liear coersio of he coer-propagaig waes. O he oher had cobiaio of his liear feare wih

2 irisic olieariy of aerials i which G ca be wrie has araced a grea deal of aeio i fdaeal ad applied research of he ligh rasissio i Geqipped srcres. Nolieariy gies rise o self-phase odlaio (SPM) ad cross-phase odlaio (XPM) ers i he copled-ode descripio of Gs. I pariclar i he case whe he G is wrie o a opical fiber he correspodig odel is based o well-kow eqaios []-[] i i i i where ad are aplides of he forward- ad backward-propagaig waes is he coordiae alog he fiber he ragg-reflecio ad XPM coefficies beig oralized o be. (.) alysis of saioary properies of his odel led o predicio of opical bisabiliy i oliear periodic srcres i 979 []. Laer a lo of work has bee doe o he heoreical []-[5] ad eperieal [6] sdy of gap solios i his syse. Gap solios owe heir eisece o he balace bewee he SPM ad XPM olieariy ad G-idced liear dispersio. The aalyical for of he solios is []-[] ( ; ψ c) sech( ξ - iψ ) ( ; ψ c) sech( ξ iψ ) ep( iθ ) ep( iθ ) ( ) ξ θ ( c ) ± c ± siψ c c c siψ c c c cosψ a c c ( co( ψ ) coh( ξ )) Solios belogig o his faily deped o he paraeers ψ ad c. The forer oe deeries he aplide widh ad ceral freqecy of he solio ad akes ales fro he ieral < ψ < π while he elociy c is liied o - < c <. Sabiliy of he solios (.) was firs cosidered by eas of a ariaioal approiaio i Ref. [7]. I was coclded ha a par of he soliary-wae faily (.) is sable. The cofiraio of he eisece of a isabiliy regio i he (c) plae of he solios (.) was laer obaied by eas of rigoros erical ehods i he works [8]-[] (see also Ref. []). The sdy of gap solios was he epaded o coposie srcres iolig Gs. The siples ersio of sch a srcre is a se of wo liearly copled parallel fibers carryig Gs. This syse was iesigaed i Ref. []. I was show ha for soe criical ale of he liear-coplig coefficie which depeds o he eergy of he solio obios syeric solios lose heir sabiliy bifrcaig io sable (.)

3 asyeric oes. Sabiliy aalysis of boh he syeric ad asyeric solio solios was perfored i Ref. [] by eas of direc silaios ad i was coclded ha whe asyeric solios eis hey are always sable while he syeric oes are always sable i he sae case; o he oher had whe asyeric solios do o eis he syeric oes are always sable. I his work we iesigae a riaglar cofigraio of hree liearly copled ideical Gs which is he os syeric possible coposie srcre cosisig of waegide graigs. I Ref. [] solios i a riaglar cofigraio of ordiary oliear opical fibers were iesigaed ad solios i a plaar raher ha riaglar ri-core cofigraio were sdied i deail i Ref. []. syse of hree waes i he spaial doai liearly copled by a riple G wrie o he srface of a plaar oliear plaar waegide was sdied i Ref. [5]. This hreewae syse which is forally aao o a odel wih ".5'' cores i he eporal doai gies rise o a rich faily of solios which coais boh reglar gap solios ad sable "cspos" ad "peakos". The cosideraio of he riaglar hree-core cofigraio of Gs is relea for arios reasos. Firs of all his cofigraio akes i possible o perfor direc swich of opical sigals bewee ay wo. Geerally he sdy of oliear dyaical saes i eqilaeral riaglar cofigraios i arios syses is a opic of fdaeal ieres for obios syery reasos see e.g. Refs. [] ad [6]. I he coe of he G syses ri-core cofigraios offer ew possibiliies o ehace fcioaliy of he graig-based deices which is a isse of cosiderable crre ieres see a reiew [7]. We deosrae ha he rasiio fro wo o hree cores drasically chages boh he liear specr ad solio coe of he syse; for isace isead of he sigle faily of asyeric solios eisig i he dal-core syse he ri-core oe gies rise o hree asyeric failies wo of which ay be sable. oher esseially ew feare of he ri-core syse is he eisece of a faily of oriial aisyeric solios (i he dal-core odel syeric ad aisyeric solio failies are aao o each oher); he sabiliy regio of he aisyeric solios akes a sal for see below. esides ha we also repor ew resls for dyaical saes i dal-core oliear graigs. esseial resl is ha he eisece ad sabiliy of syeric solios i boh he dal- ad ri-core syses coiosly eed far across he borders of he specral gap. This fidig is of direc releace o he eperie as i sggess ha he creaio of sable solios is possible i a freqecy bad ad i a paraeer regio which are ch broader ha i cold be epeced a priori. Moreoer a broad sabiliy regio for he syeric solios is fod i he case whe he gap does o eis a all hece he syse cao sppor ordiary gap solios. I addiio o ha i is fod ha he whole sabiliy regio of he aisyeric solios i he ri-core syse is locaed oside he gap (ad i also persiss whe he gap does o eis ayore). The sabiliy regio of he syeric solios is liied o relaiely sall ales of heir aplide. Howeer i os cases whe hey are sable he isabiliy which has a oscillaory characer does o desroy he solios b qickly rasfors he io robs (erically sable) breahers. The laer resl is physically relea oo

4 as i sggess a possibiliy o eperieally look for sch breahers i dal- ad ricore oliear graigs. esides ha we deosrae ha sable asyeric slighly delocalized solios (he oes wih oaishig ails) ca also be fod oside he gap which akes he ariey of sable dyaical saes aeable o he eperieal obseraio sill broader. I is relea o sress ha all hese sable saes (solios oside he gap breahers ec.) do o eis i he ordiary sigle-core graigs. s cocers heir acal realizaio dal- ad li-core fiber graigs were a heoreical cocep il he fabricaio of a dal-core waegide wih he graig syerically wrie o boh cores was repored i a ery rece eperieal work [8] (see Fig. 6 i ha paper). I ha work he dal-core fiber graig was sed as a basis for a add-drop elecoicaios filer. The sae echology shold ake he fabricaio of he ri-core syse qie feasible. Howeer he os proisig hos edi i which dal- ad ri-core graigs ay be ipleeed is a phooic crysal fiber (PCF). Ideed a gided ode ca be easily localized iside he PCF i a layer of he widh ~ arod a ceral hole of a relaiely large diaeer (~ ) see a reiew [9]. aral way o apply he G o his ode is o wrie he graig o he ieral srface of he hole. The eacly he sae odel which describes a doble- or ri-core fiber graig is also alid for he PCF ari hosig a syse of wo or hree broad holes which for a eqilaeral riagle which is qie possible echologically. Noe ha fabricaio ad applicaio of a PCF hosig wo far separaed waegidig cores was repored ery recely i Ref. [] (see Fig. i ha paper); he sae paper ephasizes ha fabricaio of arios li-core paers i he PCF ari is ch easier ha akig siilar srcres coposed of ordiary fibers. I is relea o sress ha ha alhogh a cobiaio of a PCF wih G was eioed i soe rece papers [] he possibiliy o se iered Gs wrie o he ieral srfaces of holes ad copose li-core graigs of his ype were o ye cosidered o or kowledge. cally applicaios of he iered graigs i he PCF ari ay be ch broader ha js he creaio of dal- ad ri-core graigs. I he coe of akig gap solios i fiber graigs he sregh of he olieariy is a crcial isse as he correspodig olieariy legh s be o loger ha ~ c [6]; i ordiary fibers his akes i ecessary o lach sigals whose peak power is coparable o he opical-breakdow hreshold. I he PCF seig his proble is ch easier o sole as his edi ay proide for a effecie olieariy coefficie for he gided ode p o (W k) i.e. 5 ies as srog as i he ordiary fibers []. This is aoher adaage of sig he PCF edi o creae syses spporig gap solios. To coclde he irodcio i ay be relea o eio ha aoher way o irodce a li-copoe syse i Gs is o se wo polarizaios of ligh (raher ha li-core srcres) which was deosraed eperieally i he eporal doai [] ad heoreically i he spaial oe []. The paper is orgaized as follows. The forlaio of he odel is preseed i secio II. I secio III liear properies of he srcre are cosidered. gap i he syse's specr is fod which olies he regio where gap solios are epeced o eis. Secio I preses failies of eac syeric ad aisyeric solio

5 solios (as i was eioed aboe hese solio failies eed across he gap's borders ad persis i he case whe he syse's specr has o gap a all). Secio iesigaes geeral asyeric solios. Three differe species of he (o he corary o he sigle faily of asyeric solios i he dal-core syse []) are prediced by eas of he ariaioal approiaio ad are he fod i a erical for. Secio I deals wih sabiliy of he solios. The sabiliy of asyeric solios is esed by direc erical silaios. For he aboeeioed eac syeric solios rigoros sabiliy aalysis is perfored wihi he fraework of liearized eqaios for sall perrbaios. For he prpose of his cosideraio he proble is forlaed for M syerically copled ragg graigs so ha i coprises he cases of boh wo (M ) ad hree (M ) liearly copled cores. Coclsios are preseed i secio II.. THE MODEL We sar by cosiderig M syerically copled oliear graigs where M will ake ales ad. The elecroageic field i he cores is assed i he for E ( z ) E ( z τ ) ep( ik z i τ ) E ( z τ ) ( ik z i τ ) τ ep c.c. where he sbscrip M is he ber of he correspodig core z is he propagaio disace τ is ie ad k are he freqecy ad wae ber of he carrier wae ad c.c. sads for he cople-cojgae epressio. Followig he kow echiqe of he copled-ode heory we derie eolio eqaios goerig he slow eolio of he eelope fields i he cores: E i z E i z E i τ κe E i κe τ γ ( E E ) γ E C k ( E E ) E C Ek where κ ad C are he ragg refleciiy ad he ier-core coplig cosa respeciely ad he grop elociy of ligh is se eqal o (he deriaio is a sraighforward cobiaio of hose for he sigle-core G [5] ad dal- or ri-core oliear coplers see e.g. Ref. []). E k k (.) (.) pplyig he followig oralizaios o Eqs. (.) E E κ γ κ γ z κ τ κ (.) we cas he i a ore coeie for:

6 i i i i k k k k where he sigle reaiig paraeer is C /κ. esseial differece of he ricore odel fro is dal-core coerpar is he iporace of he coplig-cosa's sig: i he case of wo copled eqaios oe ay always redefie so ha i is posiie while i he case of hree copled eqaios here is a real differece bewee posiie ad egaie ales of he coplig cosa. (.) Throgho his paper we cosider oly qiesce (zero-elociy) solios ha ca be looked for as ( ) ep( i) ( ) ep( i). Sbsiig hese epressios io Eqs. (.) leads o saioary eqaios i i ' ' k k k k where he prie sads for d/d. Forward- ad backward propagaig copoes of he qiesce solios obey he syery cosrais -. The Eq. (.6) for he -h core becoes (.5) (.6) ' i k k. (.7) elow he saioary eqaios (.7) ad eolio eqaios (.) for wo or hree cores will be sed o cosrc solio solios ad iesigae heir sabiliy. The ai sbjec will be he ew case of hree cores alhogh he wo-core odel will be reisied wo i order o obai soe sabiliy resls i a ore accrae for ha i was doe i Ref. []. Ths for he ri-core cofigraio he eplici fors of he propagaio ad saioary eqaios are respeciely

7 i i i i i i i i i i i i ad. ' ' ' i i i. THE DISPERSION RELTION ND SPECTRL GP Iesigaio of he liear specr ca proide a cle o search for a eisece rage of gap solios. Oiig oliear ers i Eqs. (.8) ad lookig for a solio i he for ( ) ep( ) ep i ik i ik we fid he followig braches of he dispersio relaio: ) ( 56 k k M ± ± which are show i Fig.. The dispersio depedeces are wrie i he for which applies o boh he dal-core (M ) ad ri-core (M ) syses; i he forer case he braches 6 5 are abse. Gap solios are epeced o eis i he specral gaps where o liear propagaio is possible. ccordig o Eqs. (.) for he ri-core syse he gap is ( ) < < for posiie ad (.8) (.9) (.) (.) (.)

8 ( ) < < for egaie while for he dal-core cofigraio he gap is <. The gap is wides whe. Wih he icrease of he gap arrows boh ierals (.) ad (.) shrikig o il a / siilar o he dal-core's gap which closes dow a. (.) I he case of wo graigs he gap is syeric relaie o he zero-deig freqecy. O he oher had i he ri-core syse his syery is abse ad he freqecy is copleely pshed o fro he specral gap for > / while he gap sill eiss i he regio / < < /. The gap regio i he paraeric space () of he ri-core syse is show i Fig.. The epressios (.) ad (.) defie he geie gap i.e. a oerlap bewee sbgaps of all he dispersio braches. Reglar gap solios are epeced o be fod iside his gap. Howeer i is kow ha i a syse wih a leas wo differe braches of he dispersio relaio he so-called ebedded solios ay eis iside oe of he sbgaps beig ebedded i he coios specr belogig o he oher dispersio brach [56]. I will be show below ha he prese syse sppors solios boh iside ad oside he geie gap. The ebedded solios are sally sei-sable (i.e. sable i he liear approiaio b oliearly sable) ad i soe cases hey r o o be i pracical ers copleely sable objecs [6].. EXCT SOLTIONS Soe soliary-wae solios of Eqs. (.6) ca be fod i a eac aalyical for. Wih respec o relaios bewee he wae fields i differe cores hese solios ay hae a syeric or aisyeric for.. Syeric solios Syeric solios hae ideical fields ad i all he cores. Lookig for he i he for ( ; ) ( ; ) ep( i) ( ; ) ( ; ) ep( i) i is sraighforward o see ha hey ca be epressed as ( ; ) ( ; ( M ) ) ( ; ) ( ; ( M ) ) (recall M is he ber of cores which akes ales or ) where (;w) ad (;w) are he sadard saioary solios (.) for he gap solios wih c ad he freqecy cosψ ( M ) i he sigle-core oliear G. These solios eis i he freqecy ieral - < cos < which is he correspodig specral gap hece he eac syeric solios ca be fod i he ieral < ( M ) < i.e. precisely i he sbgap of he dispersio braches ) defied by Eqs. (.). ( k I he case of he ri-core syse M he regio (.) is he ierior of he (.) (.) (.)

9 egaie-slope sripe i Fig.. Ths he eac syeric-solio solios eis o oly iside he gap proper b also (as ebedded solios) i he whole sbgap (.). I is also ipora o oice ha he faily of he syeric solios is prese ee i he cases > ad > / for M ad M respeciely whe he geie gap does o eis a all hece he syse sppors o reglar gap solios while he coios faily of he ebedded syeric solios is aailable.. isyeric solios The eisece of he aisyeric solio i he dal-core odel is obios []. I fac hey are aao o he syeric solios wih replaced by. I he ricore syse his syery is abse ad a aisyeric solio is a idepede solio. I has copoes wih opposie sigs i wo cores ad zero i he hird oe: for isace () ad () -() (). The eisece of his solio is possible as he liear ers geeraed by he firs ad secod fields i he eqaio for he hird core eacly cacel each oher. I he o-epy cores he saioary field for he aisyeric solio akes he for ( ; ) ( ; ) ( ; ) ( ; ) where (;) ad (;) are he sadard sigle-core solios (.) wih c ad freqecy cos cf. Eqs. (.). ccordigly he aisyeric solios eis i he ieral < < cf. Eq. (.). This regio is eacly he sbgap of he dispersio braches ad 56 fro Eqs. (.) ad i fills he sripe wih he posiie slope i Fig.. Ths as well as he syeric solios he aisyeric oes eis o oly iside he gap proper b also i he whole sbgap (he oe rasersal o ha which sppors he syeric solios). eyod he gap's borders he aisyeric solios are ebedded oes. Lasly we oice ha as well as heir syeric coerpars he aisyeric solios keep o eis (as ebedded solios) ee i he case whe he re gap is abse i he syse. (.) (.5) 5. SYMMETRIC SOLITONS: THE RITIONL PPROXIMTION ND NMERICL RESLTS I order o classify ore geeral solio solios i is aral o sar he aalysis wih he case of a sall ier-core coplig cosa (siilar o how i was doe for he dal-core odel i Ref. []). I he lii eqaios for differe cores decople ad he followig solios ca be ideified [recall () sads for he sal sigle-core solio wih he zero elociy]: Syeric wih () () () () ; isyeric () -() () () ; syeric solio ype I: () () () () ; syeric solio ype II: () () () () -() ;

10 syeric solio ype III: () () () (). I is relea o eio ha a siilar approach i which differe ypes of solios are ideified o he basis of obios fors aailable i he decopled lii is ery efficie i classificaio of arios solio-like saes i odels of dyaical laices see e.g. Ref. [7] ad refereces herei. oher esseial reark is ha i he dal-core syse he sae liiig case gies rise o a sigle ype of asyeric solios wih () () (). For he firs wo species (syeric ad aisyeric solios) eac solios alid for fiie were gie i he preios secio. To fid he saioary solios of all he ypes we firs applied he ariaioal approiaio () o Eqs. (.9); he he solios were sogh for i a direc erical for. Firs we prese aalyical resls proided by. The ri-core saioary eqaios (.9) ca be deried fro he Lagragia desiy where ( ) Q ( ) Q ( ) { } L Q Re ' ( ) I( ) Q ( ) is he Lagragia desiy for he sigle-core odel. Ne followig Ref. [] where was applied o solio solios of he dal-core odel ad yielded qie accrae resls we adop he followig asaz ( µ ) i sih( µ ) sech ( ) sech µ (5.) (5.) (5.) where he sae widh µ is assed for all he hree copoes while he aplides ad ay be differe for differe. The effecie Lagragia which is obaied by he sbsiio of he asaz (5.) i he desiy (5.) ad iegraio is displayed i ppedi. aryig he effecie Lagragia (.) wih respec o ad µ yields a syse of algebraic eqaios (.) which are also wrie i ppedi. Eqaios (.) were soled by eas of he Newo-Raphso ehod. Figres ad 5 show aplides of wo differe copoes for solios of all he ypes defied aboe as fod fro he a.5 ad.8 respeciely. Oly wo aplides are show becase he hird oe is always eqal o oe of he ad he syeric solio is represeed by a sigle cre as i has a sigle aplide. s is see fro Figs. 5 braches of asyeric solios of he ypes I ad III are geeraed fro he syeric-solio brach by pichfork bifrcaios. Frher he followig geeral coclsios ca be draw fro he diagras:

11 Syeric solios bifrcae wice: a a larger ale of io Type-I asyeric solios ad a a slighly saller io Type-III solios. The wo bifrcaio pois ed o coe closer as becoe larger. The regio of eisece of he asyeric solios of he ypes I ad III becoes saller as icreases. The ype-iii solio eiss oly iside he geie specral gap (recall i was defied aboe as he oerlap bewee he wo sbgaps). The ype-i solio also eiss iside he geie gap oly less (probably a sall fiie ales of his solio ay eis oside he geie gap). The ype-ii solio eiss boh iside ad oside he geie gap. Ne he relaaio ehod based o he so-called sic-collocaio echiqe was eployed o obai erically eac saioary solio solios sig he predicio prodced by he as a iiial gess. Iside he geie gap he ariaioal resls ach heir erical coerpars qie well which is illsraed by Fig. 6. O he oher had direc aeps o geerae asyeric solios oside of he geie gap i he erical for sarig wih he -prediced iiial gess (i hose cases whe he does predic asyeric solios oside he gap) hae failed. Isead he erical algorih prodces delocalized solios (alias "qasisolios") wih sall oaishig oscillaory ails which is a direc coseqece of he fac ha he solio does o belog o he geie gap. defiie coclsio erified by he erical ehod is ha reglar asyeric solios eis oly iside he specral gap while all he asyeric solios fod oside he gap are delocalized oes. I he laer case he ceral core of he delocalized solios is fod o be qie close o he correspodig -prediced shape i accordace wih he priciple ha ee if i isses he eisece of he ail is able o correcly describe he solio's core [8]. Typical eaples of he delocalized solio fod oside he gap are displayed i Fig. 7. epeced feare fod fro he erical resls is ha he aplide of he oaishig ail icreases as oe approaches he gap's border alog he braches of he ype-i or ype-ii delocalized solios. Hiig he gap edge he delocalized solios disappear while a reglar (rly localized) asyeric solios of he ypes I ad II show p. I fac he ceral body of he qasi-solios is described by he qie accraely. 6. STILITY fer haig fod basic ypes of saioary solio solios i he odel i is ecessary o aalyze heir sabiliy. Direc silaios of he eolio eqaios is he os cooly sed ehod for deeriig he sabiliy of solios which acally correspods o he way he physical eperie is r. Howeer ore rigoros iforaio o he sabiliy is frished by erical copaio of he correspodig eigeales wihi he fraework of liearized eqaios for sall perrbaios (for he sigle-core ragg-graig odel his was doe i Refs. [8-]). I pariclar i ay happe ha direc silaios ay soeies aifes a weak isabiliy which is a arifac of he erical schee while he solio is sable (i

12 soe cases cosidered i Ref. [] he weak isabiliy of syeric solios i he dal-core G odel was acally he arifac). elow we display resls for he sabiliy of solios i he prese odels (boh dalcore ad ri-core oes) obaied by eas of boh direc silaios ad eigeale copaio. Direc silaios were sed o iesigae he sabiliy of asyeric solios (for which erical copaio of he eigeales is a echically difficl proble) ad o delieae a sabiliy regio of syeric solios. The we applied he ore rigoros eigeale-based procedre o he syeric solios.. Direc silaios Syseaic direc silaios of he solio sabiliy hae resled i he followig coclsios for he syeric aisyeric ad asyeric solios i he ri-core syse: Syeric solios are sable before he bifrcaios occr. fer passig (i he direcio of decreasig see Figs. -5) he bifrcaio pois a which he ype-i ad ype-iii asyeric solios are bor he syeric-solio brach becoes sable. Sice wo bifrcaio pois ay be qie close i is soeies difficl o deerie eacly where he syeric solio loses is sabiliy. O he oher had sable (a leas i he liear approiaio) syeric solios are also fod oside he gap see Fig. below. The iplicaio of he laer resl for he eperie is he grea elargee of he paraeer regio (i pariclar of he correspodig freqecy bad) i which sable solios ca be sogh for. Moreoer sable syeric solios are fod i he case > / whe he gap does o eis a all i he ri-core syse. The aisyeric solio wih oe epy core is always sable iside he specral gap ad i is sable oo whe i eiss oside he gap for posiie see Figs. ad 5. Howeer a relaiely large egaie ales of he coplig cosa he solio of his ype is sable (his siaio occrs e.g. i he case show i Fig. ; recall ha like he dal-core syse i he ri-core oe he sig of is a oriial igredie). Noriial prpor of his fidig is ha sable aisyeric solios (as well as syeric oes see aboe) are possible oside he gap icldig he case whe he gap does o eis. eaple of sable eolio of sch a asyeric solio is displayed i Fig. 8 for ad -.8 (oice ha his sable aisyeric solio is fod i he case whe he re gap does o eis a all as > / i his case). The ype-iii asyeric solio is sable as i is geeraed by he bifrcaio fro he syeric solio a posiie. O he oher had he ype-i asyeric solios appear as sable oes afer he correspodig bifrcaio. Followig he braches of he ype-i ad ype-iii solios owards egaie ales of sabiliy echage bewee he was fod. For eaple he ype-iii solio is sable a ad. ad is ype-i coerpar is sable i he sae case while for ad -. he characer of he sabiliy is eacly opposie. Figre 9 displays ypical eaples of sable ype-i ad ype-iii asyeric solios. Type-II asyeric solios are always sable. Qasi-solios (delocalized oes) of he sae ype are always sable oo.

13 Type-I delocalized solios (qasi-solios) eisig oside he gap are sable for relaiely large egaie ales of. I pariclar he sabiliy was obsered a for akig ales bewee.6 ad.8. s well he sabiliy of he rly localized aisyeric solios fod oside he gap he sabiliy of he slighly delocalized asyeric oes is a predicio of sigificace o he eperie as i opes a way o look for sable paers i a ch larger paraeer regio ha i cold be epeced.. The eigeale aalysis I his paper we prese resls of copaio of he sabiliy eigeales oly for he syeric solios as i oher cases he proble is really difficl ad i shold be a sbjec for a separae work. I is relea firs o recapilae he way he eigeale aalysis was perfored for he sigle-core odel i Ref. [8]. To derie liearized eqaios for sall perrbaios arod he saioary syeric solios he perrbed solios were ake i he for ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ep ep ; ep ep ; ep ep ; ep ep ; i i i i i i i i α α α α where is he eigefreqecy (sabiliy eigeale) ad () (;) are ake as per he eac solio (.). The eigeodes of he sall perrbaios are solios o he followig liear syse ( ) σ σ α σ σ σ σ σ σ Q i where is he col coposed of σ are he Pali arices ad he ari Q is ( ) Q Eqaio (6.) ca he be rewrie as ( ) J α defiig ad J appropriaely. Eqaio (6.) ses he eigeale proble. I Ref. [8] i was show ha i he sigle-core odel here is a criical ale cr of he freqecy of he perrbed solio sch ha for > cr all he eigeales α are real [i.e. here is o isabiliy see Eqs. (6.)]. I fac cr fod i Ref. [8] is ery close o zero (irally he sae criical ale was fod earlier i Ref. [7] by eas of he ). (6.) (6.) (6.) (6.)

14 I he case of he syeric solios i he dal- or ri-core syse we asse perrbaios of he for where ˆ ( ; ) ad ˆ ( ; ) ( ˆ ( ; ) ( ) ep( iα) ) ep( i) ˆ ( ; ) ( ) ep( iα) ep i ( ˆ ( ; ) ( ) ep( iα) ) ep( i) ˆ ( ; ) ( ) ep( iα) ep i ( ) ( ) ( ) ( ) are he perrbed syeric soliary-wae solios like he oe gie by Eq. (.). Irodcig he perrbaio of he for (6.5) io Eq. (.) defiig [ ] T ad akig se of he defiiios (6.) ad (6.) we arrie a he ew eigeale proble ( ˆ ) ( ˆ ˆ ) J : α M : M J ˆ σ σ where σ is he iy ari. M M (6.5) (6.6) Sice he syse is coseraie he sabiliy ay oly be eral wih all he eigeales beig real. The acal objecie is o sole he eigeale proble (6.6) for M ad aryig he ales of ad i order o ideify a sabiliy regio i he plae () where all he eigeales are prely real. For he syeric solios Eqs. (6.6) ca be radically siplified (while for asyeric solios hey are ery ioled ha is why he eigeale aalysis is o deeloped here for he). Ideed by eas of a liear rasforaio Eqs. (6.6) for he syeric solio are cas i he for ( ˆ ˆ ) J α J ( M ) σ : J M ( ˆ ˆ ) σ ( ˆ ˆ ) : σ M i which he syse is effeciely decopled coseqely he fll se of he eigeales is he io of ses obaied by solig each parial eqaio i (6.7) separaely. The decopled eqaios ca be frher siplified akig se of he propery followig fro he defiiio (6.) (6.7)

15 ( ( ; ) ˆ ( ; ) ) σ ( ˆ ( ; ) ˆ ( ; ) ) ˆ where is a arbirary ieger. Ths we obai ( ˆ ( ; ( M ) ) ˆ ( ; ( M ) ) ( M ) ) J α ~ J ( ˆ ( ; ( M ) ) ˆ ( ; ( M ) ) ) (6.8) ~ (6.9) where ~. T cally he firs decopled eqaio iplied i (6.9) has already bee soled i he coe of he sigle-core sabiliy proble [68] which yields sable eigeales for (M-)> cr ad sable oes for (M-)< cr. Ths he syeric solios are defiiely sable i he regio < cr -(M-). The acal characer of he regio > cr -(M-) where he aboe resl does o prodce isabiliy is deeried by eigeales of he secod decopled eqaio i (6.9). Wih he defiiios ( M ) β M we arrie a a odified eigeale proble ( ( ) ( ; ) β ) αj ;. (6.) (6.) Eigeales geeraed by Eq. (6.) were fod erically sig he aboeeioed sic-collocaio echiqe wih he sic basis ageed by he sie ad cosie fcios o acco for oaishig periodic ails of he eigefcios a ± (a eesie descripio of he sic echiqes ca be fod i Ref. [9]). The copaio of he eigeales was perfored o a dese grid i he space ('β). Pois a which he syeric solios were hs fod o be copleely sable are arked i Fig. for boh he dal-core ad ri-core syses M ad (as i was eioed aboe i he dal-core syse aisyeric solios are aao o heir syeric coerpars wih replaced by - herefore i Fig. he sabiliy regios for he syeric ad aisyeric solios i he dal-core syse are irror iages of each oher). The diagoal lies wih he egaie slope show he bodary of he eisece of he syeric solios. coclsio clearly sggesed by Fig. is ha he syeric solios are sable close o he pper bodary of heir eisece regio which correspods o sall-aplide syeric solios. oher ipora iferece is ha (as i was already eioed aboe) he sabiliy regio of he syeric solios eeds far beyod he borders of he specral gap ad oreoer sable syeric solios are fod i he case whe he gap does o eis ( > for M ad > / for M ). Ths he syeric solios for a coios faily of ebedded solios [56] which are sable i he liear approiaio ee i he case whe reglar gap solios do o eis a all.

16 oher relea isse is o dersad wha will happe wih he syeric solio whe i is sbjec o he isabiliy. Or copaios clearly deosrae ha i boh he dal- ad ri-core syses he desabilizaio of he syeric solios occrs wih he icrease of he solio's aplide hrogh he eergece of a pair of cople-cojgae eigeales hece he isabiliy is oscillaory. I accordace wih ha direc silaios show ha os ypically he growh of sable perrbaios qickly saraes ad as a resl he sable saic solio rs io a breaher which feares persise irisic ibraios. To illsrae his geeric sceario i Fig. we display a ypical eaple of he desabilizaio of he syeric solio i he dal-core syse. The desabilizaio ses i a '.86 for β -. [i.e..; he paraeers are defied as per Eqs. (6.)]. Direc silaios of he sae syse displayed i Fig. cofir ha he deelope of he isabiliy rasfors he sable syeric solio io a breaher. Neerheless i rarer cases (a soe oher ales of he paraeers) he isabiliy cold copleely desroy he syeric solio. This geeric resl (he rasforaio of sable syeric solios io sable breahers) is of obios releace o he eperie as i sggess a possibiliy o obsere he breahers i he dal- ad ri-core oliear graigs. The oly ecessary codiio for he creaio of a breaher is o sar wih a sable solio haig a sfficiely large aplide i.e. lachig a plse carryig sfficiely large eergy which is o difficl [6]. Lasly we dwell o aisyeric solios i he ri-core syse. I his case he eigeale proble (6.6) ca be siplified oo. Eeally he fll sabiliy proble splis io wo decopled oes. This ie hey ake he for of ( ) σ σ ( ˆ ( ) ˆ ; ( ; ) ) ( ˆ ( ; ) ˆ ( ; ) ) αj. J α J I follows fro he separaig eqaio (6.) ha he aisyeric solio is defiiely sable i he regio < cr. The sabiliy i he reaiig regio > cr is deeried by wo copled eqaios (6.). We do o prese resls of he fll iesigaio of hese eqaios here which is a echically hard proble. Recall ha direc silaios repored i he preios sbsecio hae deosraed ha he aisyeric solios i he ri-core syse ay be sable oside he gap (or i he case whe he gap does o eis a all) a sfficiely large egaie. (6.) (6.) 7. CONCLSION I his work we hae irodced a odel of hree liearly copled oliear cores eqipped wih ragg graigs (Gs) which for a riagle. The os proisig way o creae ri-core G cofigraios (as well as dal-core oes) is o se iered graigs wrie o ieral srfaces of relaiely broad holes i he phooic-crysalfiber seig. The iesigaio of solios i his odel ad heir sabiliy is a isse of ieres i is ow righ ad offers applicaios o he desig of highly fcioal

17 phooic deices as well as o he sdy of phooic crysals cobied wih Gs. We hae also reisied he earlier sdied dal-core syse obaiig ew resls for i. Failies of syeric ad aisyeric solios were fod aalyically. They coiosly eed across borders of he specral gap i boh he dal- ad ri-core syses; oreoer hese failies persis i he case whe he syse's liear specr has o gap a all hece he syse cao sppor ordiary gap solios. par fro ha hree differe ypes of asyeric solios were fod by eas of he ariaioal approach ad erical ehods. syeric solios of all he ypes eis oly iside he specral gap b slighly delocalized asyeric solios wih oaishig ails are fod oside he gap. Sabiliy of all he solios was eplored i direc silaio of he eolio eqaios. The sabiliy of syeric solios was also sdied i a ore rigoros way by copaio of sabiliy eigeales for sall perrbaios. The resls show ha he syeric solios are sable p o bifrcaio pois a which hey gie birh o wo ypes of asyeric solios. eyod he bifrcaio oe ype of he asyeric solios is iiially sable while he oher is o. Laer hey swap heir (i)sabiliy. The hird ype of he asyeric solios is always sable. If he syeric solios are sable heir isabiliy has oscillaory characer. I os cases i does o desroy he b raher rasfors io robs breahers. I has bee fod ha he sabiliy regio of he syeric solios i boh he dalad ri-core syses eeds far beyod he specral gap; oreoer a broad sabiliy regio of he syeric solios is fod i he case (srog coplig bewee he cores) whe here is o gap i he liear specr of he syses. I addiio he whole sabiliy regio of aisyeric solios which cosie a ew ype of solios i he ri-core syse is locaed oside he gap (ad i also persiss whe he gap is abse). These fidigs sgges ha solios i he li-core Gs ca be obsered eperieally i a ch broader freqecy bad ha i heir sigle-core coerpar ad i a ch wider paraeer regio ha i cold be epeced a priori. Sable asyeric delocalized solios were also fod oside he specral gap addiioally eedig he ariey of oliear saes aeable o he eperieal obseraio i he li-core Gs. CKNOWLEDGEMENT We appreciae sefl discssios of isses cocerig he solio sabiliy wih dreas Mayer.

18 PPENDIX The effecie Lagragia prodced by he asaz (5.) is ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ). 5 5 ) ( d L µ µ µ µ µ µ µ µ aryig his effecie Lagragia wih respec o ad µ we obai a syse of algebraic eqaios ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) µ µ µ µ µ µ REFERENCES. R. Kashyap Fiber ragg graigs (cadeic Press: Sa Diego 999).. H.G. Wifl J.H. Marbrger ad E.Garire ppl. Phys. Le (979).. D.N. Chrisodolides ad R.I. Joseph Phys. Re. Le (989).... cees ad S. Wabiz Op. Le (989). 5. C.M de Serke ad J.E. Sipe Prog. Op. (99). (.) (.)

19 6..J. Eggleo R.E. Slsher C.M. de Serke P.. Krg ad J.E. Sipe Phys. Re. Le (996); C.M. de Serke.J. Eggleo ad P.. Krg J. Lighwae Techol. 5 9 (997) Maloed ad R.S. Tasgal Phys. Re. E (99) 8. I.. arasheko D.E. Peliosky E.. Zelyaaya Phys. Re. Le (998). 9.. De Rossi C. Coi ad S. Trillo Phys. Re. Le (998). I.. arasheko E.. Zelyaaya Cop. Phys. Co. 6 ().. J. Scholla R. Scheibezber.S. Koale.P. Mayer ad.. Maraddi Phys. Re. E (999).. W.C.K. Mak P.L. Ch ad.. Maloed J. Op. Soc (998).... ryak ad.. khedie J. Op. Soc.. 8 (99).... cees ad M. Saagisia Phys. Re. E 56 (997). 5. R. Grishaw.. Maloed ad G.. Gowald Phys. Re. E (). 6. M. Johasso e-pri PS/ J. Eggleo.K. hja K.S. Feder C. Headley C. Kerbage M.D. Merelsei ad J.. Rogers P. Seirzel P.S. Wesbrook ad R.S. Wideler IEEE J. Sel. Top. Qa. Elecr. 7 9 (). 8. M. Åsld L. Poladia J. Caig J ad C. M. de Serke J. Lighwae Tech. 585 (). 9. T.M. Moro ad D.J. Richardso Copes Reds Physiqe 75 ().. W.N. MacPherso J.D.C. Joes.J. Maga J.C. Kigh ad P.S.J. Rssell Op. Co. 75 ().. M.E. Poer ad R.W. Ziolkowski Op. Ep. 69 (); M.C. Parker ad S.D. Walker IEEE J. Sel. Top. Qa. Elec ()... Fiazzi T.M. Moro ad D.J. Richardso J. Op. Soc.. 7 ().. R.E. Slsher S. Spaler.J. Eggleo S. Pereira ad J.E. Sipe Op. Le ( ).... Yli D.. Skryabi ad W.J. Firh Phys. Re. E (). 5. J. Yag.. Maloed ad D.J. Kap Phys. Re. Le (999). 6..R. Chapeys.. Maloed J. Yag ad D.J. Kap Physica D 5-5 (); J. Yag.. Maloed D.J. Kap ad.r. Chapeys Maheaics ad Copers i Silaio (). 7. P.G. Kerekidis.. Maloed D.J. Frazeskakis ad.r. ishop Phys. Re. E () 8. D.J. Kap. T. Lakoba ad.. Maloed J. Op. Soc.. 99 (997).

20 9. F. Seger Nerical Mehods ased o Sic ad alyic Fcios Spriger-erlag 99 New York

21 Fig. Dispersio cres for he ri-core syse. (a) Zero or egaie ales of he ier-core coplig cosa (doed) -.5 (dashed) ad -/ (solid). (b) Zero or posiie ales of he coplig cosa: (doed).5 (dashed) ad / (solid).

22 Fig.. The regio i he paraeric plae () of he syse of hree liearly copled ragg graigs where he specral gap eiss. Two sripes are defied by Eqs. (.) ad (.). The geie gap is he rhobs fored by iersecio of he sripes.

23 Fig.. plides of differe ypes of solios fod by eas of he ariaioal ehod s. he ier-core solios i he case of. Noe ha each sybol fors wo differe cres correspodig o aplides of wo differe copoes of he solio.

24 Fig.. The sae as i Fig. b for.5. Fig. 5. The sae as i Fig. b for.8.

25 Fig. 6. Typical eaples of asyeric solios as fod by eas of he ariaioal approiaio (dashed) ad i he erical for (solid) for ad.: (a) ype I; (b) ype III.

26 Fig. 7. delocalized asyeric solio of ype I as fod by eas of he ariaioal approiaio (dashed) ad by he erical ehod (solid) for ad -.75.

27 Fig. 8. ypical eaple of he sable aisyeric solio (wih oe epy core) fod oside he specral gap. Three copoes of he solio are show i oe pael for he sake of copacess. Sall irisic oscillaios of he solio obsered i he figre are de o iiial perrbaios added o he solio.

28 a) Fig. 9. Eolio of sable asyeric solios: (a) ype I; (b) ype III. Three copoes of he solios are show i oe pael for he copacess. Sall irisic oscillaios of he solios obsered i he figre are de o iiial perrbaios. b)

29 Fig.. The sabiliy regio of syeric solios: dos ark paraeer ses () for which i was checked ha all he eigeales of sall perrbaios arod he solio are sable: (a) he dal-core syse (his pael also icldes he sabiliy regio for he aisyeric solios); ( b) he ri-core syse. The parallel lies wih he egaie ad posiie slope are eisece borders for he syeric ad aisyeric solios respeciely.

30 Fig.. geeric eaple of he appearace ad eolio of a sable eigeale i he specr of sall perrbaios arod he syeric solio (for.) i he dal-core syse. The horizoal ais is he freqecy ' of he perrbed solio redefied as per Eq. (6.) ad he absole ale of he iagiary par of he eigeale show o he erical ais is he growh rae of he oscillaory isabiliy. The ose ad sbseqe deelope of he isabiliy of syeric solios i he ri-core syse are qie siilar o hose show here ad i Fig. below.

31 Fig.. Trasforaio of a sable syeric solio io a breaher by he oscillaory isabiliy i he dal-core syse i he case of.6.. fer a rasie period he ibraios sele dow o a qasi-seady sae.