Improved Analysis of the Coupling of Optical Waves into Multimode Waveguides Using Overlap Integrals

Transcription

1 464 rogress In Elecrogneics Reserch Syosiu 5, Hngzhou, Chin, Augus -6 Iroved Anlysis of he Couling of Oicl Wves ino Muliode Wveguides Using Overl Inegrls M Sllein, C Kollec, nd G Mrozynsi Universiy of derborn, Gerny Absrc he oicl uliode inerconnecion echnology hs becoe iorn due o incresing d res in odern uli-rocessor syses In his conex, he couling of oicl wves ino uliode wveguides hs o be nlyzed Ry oicl ehods re referred, bu re no licble for ll geoeries As full-wve nlysis using he ode ching echnique is very exensive, we resen sile ehod bsed on overl inegrls, o clcule he couling efficiency of iinging oicl wves Our roch neglecs refleced wves he corresonding error is correced by lying rnsission fcor, which is h of lne wve irrdiing dielecric hlf sce We verify our roch lnr slb wveguide nd cylindricl fiber, s heir ode secr re well nown he ode ching echnique is lied s reference ehod nd he iinging wve is Gussin be wih vrying ngles of incidence nd lerl dislceens Inroducion Fuure couing hrdwre will oere wih on-chi cloc res u o GHz o rovide roorione d rnsfer res in uli-rocessor syses, n oicl inerconnecion echnology bsed on bord-inegred oicl chnnel wveguides is needed hese wveguides hve o be highly uliodl o be dble o he exising CB ss roducion [] he syse design of oicl inerconnecs requires n effecive se of design ools wih ech bsic lgorih being efficien wih resec o ie needs nd eory requireens he i of his er is o resen n effecive roch o describe he couling of oicl wves ino uliodl se index wveguide bsed on overl inegrls Invesigion of hese couling rocesses by full-wve nlysis using he ode ching echnique is very exensive, s lo of refleced nd rnsied odes hve o be regrded he ol nuber of odes needed by his ehod igh exceed few housnd hus very lrge liner syse of equions hs o be solved herefore, his ehod is very ie nd eory consuing Anoher disdvnge is h only closed srucures cn be nlyzed An ofen used roxiion is o neglec he refleced wves In his cse, evluing he boundry condiions for he rnsverse field coonens by ens of he ode orhogonliy leds o exlici exressions for he liudes of he rnsied odes involving overl inegrls he error cused by neglecing refleced wves cn be correced by lying ode-indeenden rnsission fcor he wveguide design we wn o nlyze consiss of wveguides wih recngulr cross secion As no nlyicl exressions for he odes of hese wveguides re nown, we loo wo bsic wveguides wih nown ode secr o verify our roch he firs one is lnr slb wveguide, nd he second is cylindricl fiber In our clculions, he inciden wve is Gussin be, which irrdies he wveguide inerfce wih vrying ngles of incidence nd lerl dislceens Since we use he ode ching ehod s he reference ehod, we firs give shor inroducion o his ehod he Mode Mching Mehod In order o ly he ode ching ehod discree ode secr us be rovided herefore, closed srucure wih eg erfec elecric conducing EC) wlls, lie in Fig, hs o be considered Le he rnsverse coonens of he elecric nd gneic fields in he disconinuiy lne z = be given by { E s) H s) } = C s) { e s) h s) }, wih s {i, r, } ) he uer indices i, r, nd denoe incoing, refleced, nd rnsied wves, resecively Due o he lossless erils he orhogonliy relion { } ) e s) h s) zd = Q s) δ )

2 rogress In Elecrogneics Reserch Syosiu 5, Hngzhou, Chin, Augus y E i),h i) E r),h r) E ),H ) z EC Figure : Incoing, refleced nd rnsied wves wveguide inerfce holds for ll eigenodes e nd h z is he uni vecor in xil direcion, δ is he Kronecer del, nd denoes colex conjugion Q s) corresonds o he norlized ower of ech ode As he overll cross secions in z < nd z > re of equl size, wo equivlen ses of ode ching equions cn be wrien: A c i) + c r)) = Q ) c ) Q i) c i) c r)) = A H c ), 3) Q i) c i) + c r)) = Bc ) B H c i) c r)) = Q ) c ), 4) where H denoes herie conjugion he vecors c s) consis of he liudes C s) nd he digonl rices Q s) hve he eleens Q s) A nd B re rices wih eleens defined by he overl inegrls A = e i) { h ) } ) zd, 5) B = e ) { h i) } ) zd 6) he equivlence of he forulions 3) nd 4) cn be roved by ens of series exnsion of e i) in ers of e ) 7) [] Ming use of he orhogonliy relion ) resuls in he rix equion 8) e i) = n A n e ) Q ) n 7) Q i) = BQ ) A 8) n For he exc soluion of he couling rocess, he nuber of odes on boh sides of he disconinuiy us end o infiniy Irresecive of he nuber of odes en ino ccoun, he ode ching ehod conserves he ower cross he disconinuiy in z = [] Descriion of he Aroxiion Neglecing refleced wves in 3) nd 4) leds o wo exlici exressions for he vecor c ), disinguished by he lower indices nd : Q ) c ) = Ac i), 9) Q ) c ) = B H c i), ) Boh soluions conin n unnown error following fro neglecing refleced wves hey re equivlen o series exnsion of he incoing elecric or gneic field in ers of he eigenodes of he considered wveguide In order o iniize he overll error, le us ssue h he ower rnsied hrough he disconinuiy z = cn be clculed by ) = c ) Q ) c ) ) Ming use of 9) nd ) ogeher wih 8), we cn show h, s long s he nuber of odes regrded is

3 466 rogress In Elecrogneics Reserch Syosiu 5, Hngzhou, Chin, Augus -6 sufficien, his roch conserves he colex ower hrough he disconinuiy z = : ) = c ) = c ) Q ) c ) ) B c i) 3) = c i) A Q ) B c i) }{{} 4) c8) = c i) Q i) c i) = i) 5) As we neglec refleced wves, conservion of ower is of course no our inenion Bu i gives rise o ly rnsission fcor In he nex secion we will show h his rnsission fcor equls wih good roxiion h of lne wve iinging on dielecric hlf sce wih he se refrcive index difference On he bsis of 5) he following roxiion holds for he bsolue vlues of he ode liudes: { C ) Re C ) } C), 6) wih C ) = Q ) A C i), C) = Q ) B C i) 7) Here, is he rnsission fcor of lne wve he resuls resened in he nex secion confir 6) Equions 7) fford series exnsion of he fields of he incoing wve in ers of wveguide eigenodes fored by he EC-wlls As no closed srucure is needed ny ore, he following equions cn be used insed of 7): Here E i) C ) = { E i) Q ) h ) } zd ), C = Q ) e ) { H i) } zd 8) nd H i) re he rnsverse coonens of he incoing fields in he disconinuiy z = Verificion In x, y, z ) coordines he hree diensionl) Gussin be is given by he gneic vecor oenil funcion Ax, y, z ) = Ux, y, z )ex j o z ), 9) where is uni vecor in rnsverse direcion eg x or y ), = ω ε µ is he hse consn in free sce, nd U is defined by Ux, y, z ) = A wz ) π ex x + y ) w z ex j β x + y )) ) Rz ) Φz ) ) In cse of he lnr geoery, ll field coonens re indeenden of he y-coordine hus, ll fields re eiher rnsverse elecric E) or rnsverse gneic M) nd equion ) hs o be odified [3] he gneic nd elecric field coonens re obined by H i) = A/µ nd E i) = H/jωε) Iorn roeries of Gussin be re he be rdius wz) i is hlf he be widh b), he rdius of hse curvure Rz), nd he ngle of divergence Θ Fig illusres he roeries of Gussin be nd shows he invesiged geoery he reer z is he Ryleigh rnge which is hlf he confocl reer o ge differen ngles of incidence ϑ he be xis z is urned round he y-xis In cse of he fiber, he lerl dislceen h is wo coonen vecor indicing shifs in x- nd y-direcion o ly he ehod of ode ching, he field of he Gussin be is exnded ino series of wveguide odes fored by he EC-wlls h re eiher M-odes or E-Modes h is, in cse of he lnr geoery, he roble cn be divided in M nd E his is no ossible in cse of he fiber, s he fiber odes re hybrid odes In boh cses, however, i cn be disinguished beween wo orhogonl olrizions of he Gussin be As here re no significn differences beween differen olrizions, = x is vlid in he following he rnsission fcor in 6) is hen ssued o be given by he Fresnel forul

4 rogress In Elecrogneics Reserch Syosiu 5, Hngzhou, Chin, Augus x n w z) = w ) + z z ) x z Θ ϑ n = h n n /3 z z w = β z = Rz) z + z ) z Φz) = rcn z ) w Θ = rcn z ) 3) 4) 5) Figure : Gussin be iinging he wveguide inerfce ϑ) = n sin ϑ) cosϑ) n cosϑ) n n cosϑ) + sin ϑ)), 6) wih n = n /n he resuls in Fig 3 show h his forul is suible o correc he error cused by neglecing refleced wves Fig 3 nd 3b show resuls for he lnr slb wveguide he resuls belong o core hicness of µ, disnce of 3µ of he EC-wlls o he core, nd refrcive indices of n = 56, n = 54, n 3 = 53 Shown re he couling efficiency clculed by ens of he ode ching ehod nd he relive deviion of he roxie ehod for differen roeries of he Gussin be Fig 3 shows he deendency on ϑ, nd Fig 3b he deendency on h he deviion of he roxie ehod is in ll cses less hn %, s long s here is significn couling efficiency his is lso vlid in cse of ) [%] 8 c) 5, µ 5, 5µ 4, 5µ 6 6 [%] [%] 8 [%] 4 4 5, µ 5 5 ϑ[ ] 5 6 [%] 8 [%] [%] 8 [%] 4 5,5µ 5,5µ 5,5µ 5 5,5µ 4, 5µ 4, 5µ ϑ[ ] h x [µ] Figure 3: Couling efficiency nd relive deviion for differen ngles of divergence Θ nd be widhs b = wz); ), b): slb wveguide; c), d): cylindricl fiber b) d) 5, 5µ 4, 5µ h[µ]

5 468 rogress In Elecrogneics Reserch Syosiu 5, Hngzhou, Chin, Augus -6 he cylindricl fiber Fig 3c, 3d) Here he reers re core rdius of 5µ, rdius of he EC-wll of µ, nd refrcive indices of n = 56 nd n = 54 Due o nuericl robles, n incresed error igh be inheren in he resuls for he deviion in Fig 3c, in cse of Θ = 5, b = 5µ nd lrge ngles of incidence ϑ > ) If so, hen he ower conservion 5) is no sufficienly sisfied he se is for Fig3d, wih Θ = 4, b = 5µ nd h x > µ Conclusion We hve shown h i is ossible o clcule he couling efficiency of oicl wves ino uliode se index wveguides wihou regrding refleced wves An roxiion bsed on overl inegrls nd lne wve rnsission fcor gives quie ccure resuls for significn couling efficiencies Beer resuls igh be obined wih n iroved rnsission fcor regrding n inhoogeneous hlf sce he resuls give rise o ly his ehod o wveguide srucures wih ore coliced ode secr, eg wveguide wih recngulr cross secion, s only exressions for he guided odes hve o be nown REFERENCES Schrge, J, Bierhoff, Ebedded Oicl Wveguides for On-bord Inerconnecions, Froniers in Oics, 87h OSA Annul Meeing, ucson, Arizon, USA, Ocober 3 Elefherides, G, A Or, L Kehi, G Rebeiz, Soe Iorn roeries of Wveguide Juncion Scering Mrices in he Conex of he Mode Mching echnique, IEEE rns Microwve heory ech, Vol 4, , Ocober Sllein, M, C Kollec nd G Mrozynsi, Couling of Gussin Be ino lnr Slb Wveguide using he Mode Mching Mehod, Exended ers of he rogress in Elecrogneics Reserch Syosiu IERS 4), 39-3, is, Ily, Mrch 4