Numerical Analysis of Modal Noise in Optical Waveguides

Transcription

1 Numeicl Anlysis f Mdl Nise in Opticl Wveguides GYEONG-L KWEON Optelectnic Device L. LG Cle Ltd., Hgye-dng, Dngn-gu, Anyng-shi, Kyungki-d REPUBLC OF KOREA Astct: - Using the semi-nlytic tnsmissin nd ppgtin mtices f guided pticl wves, the mdl nise in pticl wveguide is investigted f simple thee-sl wveguide nd step-index singlemde fie. t is shwn tht mdl nise cn ise fm the unund dtin mdes which hve een excited in the fist unctin nd hven t een sufficiently ttenuted efe eching the secnd unctin. Cnsequently, ensuing singlemde is nt enugh even if ll cldding mdes e ls suppessed. t is suggested tht ll the ditin mdes shuld e ls suppessed t eliminte the mdl nise nd its detimentl effect n the pticl netwks. The mdl nise due t guided mdes is, hweve, sevee thn tht medited y ditin mdes. Key-Wds: - Mdl Nise, Mdl ntefeence, Fie Splicing, Wveguide Discntinuity ntductin t is genelly ccepted tht ut tw metes f spcing etween tw cnsecutive splicing pints in pticl fie netwk shuld e mintined t keep the mdl nise elw n cceptle limit. Hweve, the wide sped deplyment f cmplex pticl netwks nd eve incesing pcking density mkes it difficult t fllw this guideline. Als, in the tempy epi splices emegency esttin, in n pticl device pckging such s lse pigtiling, the fie discntinuity cn ften e clse thn they e desied. The pesence f mdl nise which is unvidle in such cses cn seveely degde the pefmnce f high it te systems. Theefe it will e dvntgeus t get cle undestnding f the mechnism nd find slutin t minimize the mdl nise even f the clsely spced splicing pints. A simple expnntin f the mdl nise in fie ptic netwk is s fllws. t is sid tht t the fist splice pint, the LP mde is excited lng with the LP mde. The LP mde hs highe ttenutin cnstnt thn the LP mde. When the secnd splicing pint is clse t the fist ne, hweve, the LP mde is nt sufficiently ttenuted nd the LP nd LP mde intefee t the secnd splicing pint. Since the suce wvelength in fie ptic netwk is nt ctively cntlled, its wvelength cn dift y mny intenl nd extenl fcts, such s device ging, mient temepetue fluctutin, is cuent chnge nd etc. The chnge in utput wvelength mkes diffeence in the excittin efficiency f the LP nd LP mde t the splice pint, nd t mke mttes wse, LP nd LP mdes f diffeent wvelength hve diffeent ppgtin cnstnts in the fie. The pssility f the chnge in the distnce etween the tw splice pints itself cnnt e excluded, f exmple y mient tempetue pessue fluctutins. All these cntiute t ende the time fluctuting utput pwe in the utput fie. Theefe t keep the mdl nise lw, it is ette t mintin the minimum fie length etween the splices t keep the cutff wvelength s lw s pssile[]. Hweve, K. Ae nd cwkes shwed tht the mdl nise pesist even f ve the cutff, nd went ff nly when the fie stu ws immesed in n index mtching il. Theefe they cncluded tht the cldding mde must e stipped ff t eliminte the mdl nise[]. Even in tht cse, thugh, the mdl nise is nt cmpletely eliminted s cn e esily checked y immesing the sht fie stu in index mtching il. We elieve the unund ditin mde which hs een excited t the upt tnsitin pint is nt eliminted duing the ppgtin in the sht fie stu wveguide sectin, nd this cn significntly cntiute t the seved mdl nise ehvi. n this ticle, the ppsitin is investigted using numeicl nlysis sed n the semi-nlyticl lge f the guided pticl wve ppgtin css upt wveguide unctins nd cnstnt wveguide sectins.

2 Theeticl Bckgund n this sectin, fmlism t clculte the mdl nise in pticl wveguide is develped which is ccute within the limit f scl wve ppximtin nd numeicl diffeencing e[-]. The genel pcedues e () Otin finite-diffeence slutin f the Mxwell s equtin in the tnsvese diectin while mtin in the lngitudinl diectin is teted nlyticlly. () Secue thgnlity f the slutin y using ecipcity theem. () Nmlize the slutin f unit enegy flux. () Using the tnsmissin nd ppgtin mtices, tin the ttl tnsmitted electic fields css the middle pticl wveguide sectin. () Expnd the tnsmitted field in the sis f lcl thnml mdes. () The wvelength dependence f the expnsin cefficients cespnd t the mdl nise. We cnfine u discussin t TE mdes, extensin t TM mde is stightfwd. Suppse we hve lssless nn-mgnetic pticl wveguide with cnstnt css-sectin lng the ppgtin diectin z. The scl wve equtin f TE mde is given s E () + k ε ()() E () whee ε () is the el eltive dielectic pemittivity f the stuctue, E () the cmplex electic field mplitude nd k is the fee-spce ppgtin cnstnt. The finite diffeencing f the equtin esults in mtix eigenvlue equtin given y[] d E dz [ A] E λe () whee [ A ] is the chcteistic mtix f the wveguide, λ is the eigenvlue, nd E is the clumn vect with N cmpnents cespnding t the nume f discetiztin pints. f unifm diffeencing scheme in Ctesin cdinte is tken, then the mtix [ A ] is el symmetic f lssless wveguide, nd the cespnding eigenvects e thgnl t ech the. When the eigenmde is enmlized y the sque t f the ppgtin ( ν) ( ν) cnstnt f the mde (i.e. E E β ν ), the electic field is effectively nmlized with unit pwe except f cnstnt multiplictive cnstnt. ( ν) ( µ )* µ dxdye E δ ν µ β, () whee β µ is the ppgtin cnstnt f mde µ, nd ( µ ) E is the cespnding tnsvese cmpnent f the electic field vect. δ stnds f the Cnecke s delt nd the steisk indictes the cmplex cnugte. The fml slutin f eq. () is given s[] E i [ A]( z z ) () z e E( z ) () whee z is sme efeence psitin within the sme wveguide nd i is the imginy nume. The sque t mtix [ A] is given s [ ] [ U ][ Λ][ U ] + A () whee [ U ] is the mtix whse ith clumn is the eigenvect f the chcteistic mtix [ A ] with eigenvlue λ i, [ U ] + the dint f [ U ], nd [ Λ ] is dignl mtix s given y [ ] i, λ δi, β δi, Λ () Next we need tnsmissin nd ppgtin mtices. Suppse we hve tw wveguides A nd B in cntct t z, nd th f the wveguides hve cnstnt css-sectins lng the ppgtin diectin z. We futhe ssume tht pt f the electmgnetic wve ppgting in the psitive z diectin in the wveguide A is eflected t the intefce nd the est is tnsmitted css the wveguide unctin. Bth f the eflected nd tnsmitted wves my cntin highe de mdes. Let the chcteistic mtix e [ A ] nd [ B ], espectively. At the wveguide unctin, the incident electic field is cnveted int vius mdes f tnsmitted nd eflected fields whee the tnsmissin mtix f the TE mde is given y [] [] [ A] [ A] [ B] T BA + while the eflectin mtix is given y ()

3 [ ] [ A] [ B] [ A] [ B] R ABA () + whee [ T ] BA is the tnsmissin mtix fm wveguide A t wveguide B, nd [ R ] ABA is the eflectin mtix fm the side f wveguide A. On the thehnd, ppgtin ve the distnce z f the wveguide B cn e epesented y single mtix given y i [ B] z [ ( z )] e P (9) B The mdl nise cn e nlyzed y ssuming thee wveguides s illustted in Fig.. F simplicity, the fist nd thid wveguides e ssummed s identicl in stuctue. The fmlism is, hweve, eqully vlid f ity set f wveguides. W y n z n y A B C(A) Fig.. Thee thee-sl wveguide with ltel mislignment. The ttl cmputtinl windw size is W, The ce size d, ltel ffset y, nd the ce nd cld efctive indexes e n nd n espectively. n the digm, the fist nd thid wveguides e ssumed s identicl in stuctue. The ttl tnsmitted field fte the secnd unctin is pduct f tnsmissin [ T ] BA, ppgtin [ P B ( z )], nd secnd tnsmissin [ T ] AB. This field cn e decmpsed int eigenmdes f the wveguide C(A) s given y ( ν) ( µ ) [] T [ P ( z )][ T ] E E AB B BA A µ A () µ d lg µ L s () The wvelength dependence f the cupling lss is the mdl nise. Multiple eflectins within the middle wveguide(b) is usully negligile f the cses f pcticl inteest. Even f fie-i intefce, the eflectnce is less thn %, nd p splice pint hs much smlle eflectnce. Mdl Nise in Pln Wveguide Numeiclly, pln wveguide is much simple thn pticl fies, nd cnsequently u fmlism is tested f pln wveguide fist. As is well knwn, TE TM mde is n exct slutin f Mxwell s equtins in pln wveguides. Theefe the pevius fmlism pplied t the TE mdes f pln wveguide is ccute within the limit f discetiztin e. The TE mdes f pln wveguide stisfy the fllwing equtin. E z ( ν ) x ( y,z) ( ν β E ) ( y,z) Ex y ( ν ) ( y,z) ν + k ε x ( ) ( ν y E ) ( y,z) x () The mdl nise cn e nlyzed fllwing the pcedue utlined in the peceding sectin. As n exmple, thee cnsecutive thee-sl-wveguides e mdeled. The thee wveguides e identicl in size, nd the efctive index is mdeled fte the silicn-silicn wveguide. The ce size f µm mde sue tht the guide is stictly singlemded ve. µm. The middle guide hs finite ltel ffset t induce splicing lsses nd pssile excittin f the highe de mdes. Fig. shws the effective efctive indices f the thee sl wveguide. t clely indictes tht the wveguide hs tw und mdes elw µm nd nly ne ve. Theefe y widesped elief, the mdl nise shuld exist nly unde µm, nd nne ve. Theefe the expnsin cefficient µ cn e tined using the thnmlity f the slutin nd the splicing lss L S in db is given s

4 effective efctive index wvelength (µm) Fig.. The effective efctive indexes f the thee sl wveguide. The ce width is µm, nd the ce nd cld efctive indexes e. nd., espectively. Ttl cmputtinl windw is µm nd the discetiztin pint is. Cuve is the effective efctive index f the fundmentl mde nd cuve is tht f the fist de mde. This indictes the degee f excittin in the secnd wveguide y the fundmentl mde f wveguide A. cupling lss (db) wvelength (µm) Fig.. The mdl nise. Cuve is f the wveguide B with µm length, while cuve is f length f mm. All the the pmetes e the sme s in Figs.(-). splicing lss (db) wvelength (µm) Fig.. Splicing lsses etween the und mdes f the wveguide A nd B. The ltel ffset is.µm. Cuve is the splicing lss etween the fundmentl mde f wveguides A nd B, while cuve is the splicing lss etween the fundmentl mde f wveguide A nd the fist de mde f wveguide B. Fig. cnsides the fist int nly. The ltel ffset is hlf the size f ce, i.e..µm. Cuve shws the splicing lss etween the fundmentl mdes f wveguide A nd B, while cuve shws the splicing lss etween the fundmentl mde f wveguide A nd the fist de mde f wveguide B. cupling lss (db) c wvelength (µm) Fig.. The mdl nise s functin f the ltel ffset. The middle wveguide(b) length is fixed t mm. The ltel ffset is %(cuve ), %(cuve ), nd %(cuve c) f the ce size. Fig. shw the mdl nise. Cuve is f the middle wveguide length f µm. The cupling lss is mntniclly decesing functin f wvelength nd we cn ssume tht the mdl nise penlty f this cse will nt e lge, ecuse smll chnge f suce wvelength cuses smll chnge f

5 utput pwe nly. Cuve is f the middle wveguide sectin f mm length. The mdl nise stts t egin significnt, nd the suce wvelength chnge will induce ppecile munt f utput chnge. F the lnge length f the middle wveguide, the cuve ecmes vey wiggly. Nte tht the mdl nise is pesent even ve the cutff wvelength, lthugh the scilltin mplitude hs decesed cmped t tht elw the cutff wvelength. n this exmple, the fundmentl mde is cupled t the ditin mde, nd it cuples ck int the nly guided mde f the wveguide. Fig. shws the mdl nise s functin f the ltel ffset f the wveguide B. As is cle fm the figue, the cupling lss is genelly smlle f the smll ltel ffset, nd the wvelength dependence is ls smlle f the smll ltel ffset. Mdl Nise in Opticl Fie Stu Mdl nise is intiguing plem in fie ptic netwk, nd it is well wth t nlyze mdl nise ising fm sht pticl fie stu. Me elistic cse will e the ne whee tw p splicing pints e eltively clse t ech the. The middle fie stu will e usully identicl t the dint fies, nd the mdl nise ises when the p splicing cnnectin hs intduced ltel ffset which is ig enugh t excite highe de mdes. This cse is difficult t nlyze, hweve, f it will equie enmusly lge mtices nd cmputtinl time. nsted, we will cnside the cse whee middle fie is diffeent fm the dcent fie, nd they e pepectly ligned. n u exmple, sht stu f 9 fie is inseted etween tw stndd singlemde fies withut ny ltel ffset. Theefe it is impssile t excite LP mde n tw esns. Fist, the fie stu is f ve the cutff wvelength when peted ne. µm. Secndly, thee is n ltel ffset, nd theefe y the symmety, LP mde cn nt e excited even if LP mde existed. F wekly guided pticl fies, scl wve ppximtin is sufficiently ccute f pcticl pupses. The scl wve equtin f the TE mde in cylindicl cdinte is given y E z (,z) A () E(,z) k " ε () + + E(,z) () whee the eltive dielectic pemittivity is ssumed s ttinlly symmetic, nd the intege " detemines the ngul dependence f the field s i" φ E (, φ,z) E(,z) e. The cmputtinl windw f size mx is discetized with L mx pints. The discetiztin scheme used in this study is chsen such tht L L L mx mx () Since the fields t the tw utemst pints e detemined y the undy cnditin, the esulting mtix [ A] hs the size f mx mx, whee mx L mx - nd L -. The tidignl elements f the chcteistic mtix [A] e given y [ A] [ A] [ ],, + k " ε + ( )( ) ( + + ) ( )( ) + + () () ( + ) ( )( ) A, () + we estict u inteest t the ttinlly symmetic mde with ", since we will cnside the cses whee t lest ne f the tw fies is singlemde fie, nd mdes with diffeent ngul mde nume " des nt mix when the ltel nd ngul ffsets e ze. Then simple Neumnn undy cnditin cn e used. The symmetizing mtix [ D ] f cylindicl cdinte cn e chsen s [ ] δ ( ) D (),J,J + This mtix [ D ] is equied t tnsfm the mtix [ A ] el symmetic[]. Othewise, the sque t mtix cn nt e tined in genel. Fig. shws the efctive index pfiles f tw typicl pticl fies nd thei lwest de eigenmdes. Cuve is the stndd singlemde

6 pticl fie (std SMF), cuve is typicl 9 fie. Cuves c nd d cespnd t cuves nd, espectively. Cmputtinl windw is µm nd the nume f discetiztin pint mx is. All the the pmetes e the sme s in Fig.. efctive index.... c d..... field pfile Fig. shws the mdl nise s functin f middle fie stu length. As cn e seen, cupling lss t the shte wvelength side is vey sensitive t the fie stu length, while cupling lss t the lnge wvelength side is eltively stle ginst the chnge in fie stu length dil distnce (µm) Fig.. Tw typicl pticl fies nd thei fundmentl mdes. Cuve is the efctive index pfile f stndd singlemde fie with ce efctive index n.9, nd cld efctive index n.. The cld efctive index is cmmn f the tw fies. The ce dius is. µm. Cuve is the efctive index pfile f typicl 9 fie with n. nd ce dius. µm. The wvelength dependence f the efctive indexes e neglected. The eigenmdes pltted e the fundmentl mdes f the espective fies. Cuves c nd d cespnd t cuves nd, espectively. The wvelength f light is tken s. µm, nd the nume f discetiztin pint mx is 9. cupling lss (db) 9 fie stu length (µm) Fig.. Cupling lss s functin f fie stu length t.µm (cutve ) nd. µm (cuve ). All the the pmetes e the sme s in Fig..... cupling lss (db) d c cupling lss (db) wvelength (µm) wvelength (µm) Fig.. The mdl nise s functin f fie stu length. Cuves,, c nd d cespnd t µm, µm, µm, µm f the fie stu length, espectively. Fig.9. Cupling lss s functin f wvelength f middle fie stu length f mm. All the the pmetes e the sme s in Fig..

7 Fig. shws the cupling lss s functin f middle fie stu length t tw epesenttive wvelength, ne elw cutff (cuve ), nd the the f ve cutff (cuve ). The cupling lss t. µm shws highly scillting ehvi t sht fie stu length, ptilly expling the ehvi in Fig.. Fig.9 shws the mdl nise s functin f wvelength. The middle fie stu length is fixed t mm. The mdl nise is pesent ve nd elw the cutff wvelength. Hweve, the mdl nise is me pnunced elw the cutff wvength. All the nlysis e sed n numeicl expeiment n pticul cnfigutins. Theefe, n genel cnclusin cn e dwn with cetinty. Hweve, it stngly suggest tht mdl nise cn ise y ditin mde s well s und mdes. Wveguide ntefces", J. Lightwve Technl.,, - (99) [] G. Kwen nd. Pk, "Splicing Lsses Between Dissimil Opticl Wveguides", J. Lightwve Technl.,, 9- (999). [] D. Schultz, C. Glingene, nd E. Vges, "Nvel genelized finite-diffeence em ppgtin methd", EEE J. Quntum Electn.,, - (99). [] -H. Tn, G. L. Snide, L. D. Chng, nd E. L. Hu, "A self-cnsistent slutin f Schödinge-Pissn equtins using nnunifm mesh", J. Appl. Phys.,, - (99). Cnclusin n this ticle, the mdl nise is numeiclly nlyzed f simple pln pticl wveguides nd pticl fies. We hve shwn tht mdl nise is pesent even whee the middle wveguide is stictly singlemded, nd theefe the mdl nise must hve isen medited y ditin mdes which hve een excited t the fist unctin nd hven t een sufficiently ttenuted efe eching the secnd unctin. Hweve, the mdl nise ehvi is pnunced when thee e me thn ne guided mde in the middle wveguide. Bsed n these nlysis, it is suggested tht ll the highe de mde shuld e sufficiently ttenuted t cmpletely eliminte the mdl nise in fie stus. The degee f necessy ttenutin will depend n the intended size f the fie stus. Efft t ficte mdl-nise-fee pticl fie is unde pgess in u lty. Refeences: [] S. Heckmnn, "Mdl Nise in Single-mde Fies Opeted Slightly ve Cutff", Electn. Lett.,, p. 99- (9) [] K. Ae, Y. Lcix, L. Bnnell, nd Z. Jkuczyk, "Mdl ntefeence in Sht Fie Sectin: Fie Length, Splice Lss, Cutff, nd Wvelength Dependences", J. Lightwve Technl.,, - (99) [] G. Kwen,. Pk, nd J. Shim, "A Cmputtinl Methd f Detemining Reflectnce t Aupt