Computation of the Filtering Properties of Photonic Crystal Waveguide Discontinuities Using the Mode Matching Method

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1 World Acdy of Scnc, Engnrng nd chnology Inrnonl Journl of Elcroncs nd Councon Engnrng Copuon of h Flrng Proprs of Phoonc Crysl Wvgud Dsconnus Usng h Mod Mchng Mhod Ahnsos hohrds, hos Klks, Ionns Nokosds, nd hos Sphcopoulos Inrnonl Scnc Indx, Elcroncs nd Councon Engnrng ws.org/publcon/3965 Absrc In hs ppr, h pplcon of h Mod Mchng (MM) hod n h cs of phoonc crysl wvgud dsconnus s prsnd. h srucur undr consdron s dvdd no nubr of clls, whch suppors nubr of gudd nd vnscn ods. hs ods cn b clculd nurclly by n lrnv forulon of h pln wv xpnson hod for ch frquncy. A rx quon s hn ford rlng h odl pluds h bgnnng nd h nd of h srucur. h hory s hghly ffcn nd ccur nd cn b ppld o sudy h rnssson snsvy of phoonc crysl dvcs du o fbrcon olrncs. h ccurcy of h MM hod s coprd o h Fn Dffrnc Frquncy Don (FDFD) nd h Adjon Vrbl Mhod (AVM) nd good grn s obsrvd. Kywords Opcl Councons, Ingrd Opcs, Phoonc Crysls, Opcl Wvgud Dsconnus. P I. INRODUCION HOONIC Crysls (PCs) [],[2] r consnly rcng ncrsd non s ponl soluon for h rlon of ulr-copc ngrd opcl crcus. h srong confnn of lgh n PC wvgud (PCW) llows h dsgn of shrp wvgud bnds n whch lgh cn chng drcon 90 o whou sgnfcn powr losss [3]. hs s n conrs o convnonl low ndx-conrs ngrd opcl coponns n whch h bndng rd us b kp rhr lrg (n ordr o l h bndng losss). Lrg bndng rd y ncrs h ovrll s of h ngrd crcu such s h Arryd Wvgud Grng [4]. PC-bsd dvcs cn prfor uch phoonc funconly such s lgh gnron [5] nd procssng [6]. Vrous dsgns hv lso bn donsrd for opcl flrng [7]-[]. Mny of h fornond dsgns r bsd on h nroducon of dsconnus (dfcs) nsd PCW. Fn Dffrnc Don (FDD) [2] s n ccur hod for sudyng lcrognc probls ncludng h sulon of ny PC-bsd dvcs. h n drwbck of hs hod s h s h dvc undr consdron ncrss n s h hod dnds ncrsd ory rsourcs nd Mnuscrp rcvd Spbr 30, h uhors r wh h Opcl Councon Lborory of h Nonl nd Kpodsrn Unvrsy of Ahns, Pnpsopols Ilyss GR5784, Ahns Grc (hoh@d.uo.gr). hs work ws suppord by h PENED2003 progr of h Grk Scrr for Rsrch nd chnology. copuonl. Espclly n h cs of PCW, long Prfcly Mchd Lyr (PML) scons us b usd n h npu nd oupu of h dvc n ordr o prvn rflcons [3]. In ddon, h s of h grd cn pos rsrcon n odllng sll dnsonl flucuons du o fbrcon prfcons. On h ohr hnd h Fn Dffrnc Frquncy Don (FDFD) hod [4] could b odfd n ordr o ccoun for sll gory prurbons [5] bu rqurs prohbvly lrg ory rsourcs n ordr o solv prccl PC bsd dvc probl. In hs ppr, w donsr h ffcvnss of n lrnv hod bsd on Mod Mchng (MM) [6] n h nlyss of PCW dsconnus, such s h ons ncounrd n PC-bsd flrs. I s shown h h MM hod cn provd ccur rsuls, vn for sll vrons of h gorcl prrs, whou rqurng sgnfcn ory rsourcs nd copuonl. In h frwork of h MM hod, whnvr dsconnuy s ncounrd nsd wvgud, vnscn ods r xcd nd w p o ch h gudd od n h PCW wh h odl flds n h dsconnuy cll. hs llows h copuon of h rflcon nd rnssson coffcns of ch gudd wvgud od. In hs wy, h MM hod provds usful physcl nsgh o h probl. Furhror snc n os lrg dvc dsgns, dsnc cll yps of dsconnus r ncounrd, on nds o clcul h odl flds only onc for ch yp of cll. hs cn sgnfcnly spd up h copuon procss. In prcc PC-bsd dvcs xhb prfcons du o fbrcon olrncs. I s porn o b bl o sudy hs dvcs for sll vrons fro h horcl dvc prrs. Fn Dffrnc hods nd hug ory rsourcs o k no ccoun such sll vrons. Adjon Vrbl Mhod (AVM) [5] s n ccur hod o drn h snsvy of funcon, for xpl h rflcvy of dvc, wh rspc o ny dsgn prr. In hs ppr w show h h MM hod s hghly ccur vn for sll vrons of h dsconnus n PCW nd s rsuls r coprd o AVM. h rs of h ppr s orgnd s follows: n scon II h odl flds of h PCW nd h dsconnuy cll r copud for boh gudd nd vnscn ods gvn frquncy ω. In scon III, h MM hod s prsnd n h Inrnonl Scholrly nd Scnfc Rsrch & Innovon 2(6)

2 World Acdy of Scnc, Engnrng nd chnology Inrnonl Journl of Elcroncs nd Councon Engnrng cs of PCW dsconnus nd n scon V, h rsuls of h MM hod r coprd o so xpl css wh h FDFD nd AVM. ˆ 0 B = 0 (6) Inrnonl Scnc Indx, Elcroncs nd Councon Engnrng ws.org/publcon/3965 II. CALCULAION OF HE MODES In ordr o pln h MM hod, on frs nds o clcul h gudd nd vnscn ods suppord by ch cll of h srucur nd hr propgon consns β. In gnrl, h vnscn ods of prodc srucur y hv coplx β [] nd hnc, hr propgon consns y l on h nr coplx pln nd no jus on h rl or gnry xs s n consn cross-scon wvguds. Hnc convnonl pln wv xpnson hod, whr ω s clculd for ch propgon consn β s no subl snc s uch or prfrbl o b bl o drn h vlus of β=β(ω) corrspondng o gvn frquncy ω. In hs scon, hod for drnng h vnscn nd gudd od proprs of prodc srucur gvn ω, s oulnd bsd on h forulon of h sourc-fr Mxwll s quons n rs of gnrld Hrn gnprobl s n Rf. [7],[8]. hs hod wll b usd n ordr o clcul h odl flds rqurd n ordr o pln h MM hod llusrd n h nx scon. Usng Bloch s hor, h ods of prodc dlcrc srucur long h -drcon cn b wrn s n Rf. [2] Er () = ur () jβ ( Hr () = vr () jβ (2) whr β s h propgon consn of h od nd u,v r prodc funcons long h drcon. Dfnng β> o b four coponn vcor coprsng of h ngnl prs u nd v of u nd v rspcfully,.. ( ) ( ux uy vx vy) β >= u, v =,,, (3) on cn wr Mxwll s quons n h followng for [9] ˆ ˆ A j B β βbˆ + >= β > whr h oprors  nd ˆB r dfnd by ωε 0 Aˆ ω μ = (5) 0 ωμ ω ε nd (4) In (5), ε nd μ r h dlcrc consn nd h gnc prbly of h srucur. h gnvlus of h gnprobl (4) cn b usd o drn h propgon consns of boh vnscn nd gudd ods of h srucurs whl h gnvcors drn hr odl flds. In ordr o solv (5) on cn xpnd h prodc four coponn vcor n rs of pln nd sndng wvs s Gxx Gny y jgl β >= BG ( nl )sn sn (7) > 0, n> 0, l 2 2 whr B(G nl ) r h Fourr coffcns of β> nd w hv ssud h h prodc cll s rcngulr nd n hs cs G nl =[G x,g ny,g l ] whr G x =2π/b, G ny =2πn/d, G l =2πl/ nd b,d, r h ss of h cll long h x,y nd drcon rspcvly. No h h flds n (7) vnsh h dgs of h cll nd hnc h srucur cn b hough s bng nclosd by prfcly conducng wlls. hs suon s rnscn o h rguns usd o obn h rdon od spcru of spl slb wvgud [20]. In hory b nd d us b kn nfn bu slrly o h cs of h dlcrc slb wvgud, s h wlls ov furhr nd furhr pr fro h wvgud cnr, h gudd ods of h wvgud rn prcclly h s whl or vnscn ods nd o ppr hvng hr fld prrly ousd h cor of h PCW. hs ns h for dsconnus nr h cor hs ods wll no b sgnfcnly xcd nd hnc wll no ffc h rnssson nd rflcon of h gudd ods. Hnc n prcc b nd d r ssud fn nd hr vlu us b kn so h h gudd ods of h srucur dcy sgnfcnly nr h prfcly conducng wlls of h cll. For 2D PCW whr ε dos no chng wh y, h gnprobl s furhr splfd n h rnsvrs Mgnc (M y ) cs, snc on nds o consdr only on y-drcd lcrc nd on x-drcd gnc fld ngnl coponns whch w wll dsgn s u y nd v x. In hs cs h flds do no dpnd on y nd hnc h rcprocl lc vcors G nl r such h G nl = G l =[G x,0,g l ]. Subsung (7) n (4), nd fr so hcl npulon h opror gnprobl s rnsford o rx gnprobl whch s wrn s whr h vcor MV = βv (8) ( V,..., V, U,..., U ) V = (9) N N Inrnonl Scholrly nd Scnfc Rsrch & Innovon 2(6)

3 World Acdy of Scnc, Engnrng nd chnology Inrnonl Journl of Elcroncs nd Councon Engnrng Inrnonl Scnc Indx, Elcroncs nd Councon Engnrng ws.org/publcon/3965 coprss of ll h spcrl coponns V,,V N nd U,,U N of v x nd u y rspcfully (no h fn nubr N of spcrl coponns us b ssud for copuonl purposs). h (2N) (2N) squr rx M dpnds on ω nd conns ll nforon of h dlcrc consn xpndd n Fourr srs. Fg. A srucur coprsng of dsconnus wh rbrrly posond dlcrc rods bwn wo PCW clls III. MODE MACHING MEHOD In hs scon h quons rld o h MM hod r drvd. Fg. dpcs h gnrl suon whr squnc of N clls connng dlcrc rods s consdrd. h fld us ssfy h connuy quons,.. h ngnl flds h lf of boundry us qul h ngnl flds h rgh of h boundry. A h h cll h ngnl lcrc nd gnc flds r wrn s: ( ) β ( ) ( ) ( ) () () jβ () () j = + b () (), () nd E (0) whr β r h ngnl lcrc Bloch funcons, h coffcns nd h propgon consns of h h forwrd od of h h () cll rspcvly, whl nd () b r h ngnl lcrc Bloch funcons propgon consns of h h bckwrd od of h h cll rspcvly nd ( ) β ( ) ( ) ( ) () () jβ () () j = + b () () h, nd H h h ( whr h r h ngnl gnc Bloch funcons propgon consns of h h forwrd od nd h h bckwrd od h h cll rspcvly. A ch nrfc bwn wo clls, h ngnl flds us b connuous nd h boundry = [2], hs pls: E E (2) H ( ) H ( ) (3) + ( ) = ( ) + = kng h ngrl of h lcrc fld n (2) ulpld by h gnc Bloch funcon of h (+ h ( + cll, h n, nd h ngrl of h gnc fld n (3) ulpld by h lcrc Bloch funcon of h h cll, () n rspcvly for n M ylds rx quon rlng h od coffcns n clls nd +: + A A + =Z B B (4) whr vcors A =[,, M ], B =[b,b M ] conn h coffcns of h M forwrd nd M bckwrd ods for h cll. h rx Z s gvn by Z = Y X (5) whr h ln of h rcs Y nd X r gvn by ( ) () ( + jβ, hn n, M () ( +, hn, = n ( ) () () jβ h,, n [ X ] M n M (6) n M M () () M + n, 2M h, n And ( + ( +, hn n, M ( + ( + ( + jβ, hn M, n M [ Y ] = n (7) ( + ( ) h, n, M M n ( + ( + ( ) jβ, 2 M + n M h, n If h srucur consss of ny clls, h on cn rl h odl pluds h npu o h odl pluds of h oupu usng h followng quon: nd h rx Z s gvn by N A A =Z N B B N (8) Z = Z... Z (9) o xn h rnssson nd rflcon proprs of h Inrnonl Scholrly nd Scnfc Rsrch & Innovon 2(6)

4 World Acdy of Scnc, Engnrng nd chnology Inrnonl Journl of Elcroncs nd Councon Engnrng Inrnonl Scnc Indx, Elcroncs nd Councon Engnrng ws.org/publcon/3965 srucur on cn s ll h oupu bckwrd ods qul o ro nd ssu h only h gudd ods r xcd h npu. In hs cs on obns B = Z Z A (20) A A B ( N = Z + Z2 whr h rcs M M subrcs of Z r drnd by Z Z Z 2 = Z2 Z 22 (22) o sur, n ordr o clcul h rnssson nd rflcon proprs of srucur on cn dvd no N scons nd clcul h rcs Z ch boundry. On cn hn obn h Z rx usng (9) nd clcul h pluds of h coffcns of h bckwrd ods h npu usng (20). h odl pluds of h forwrd ods h dvc oupu r gvn by (2. On cn hrfor s h rnssson nd rflcon coffcns of h gudd ods hrough h srucur. Rfrrng o fgur, no h h bgnnng of h dvc = 0, on cn ssu h h PCW clls xnd nfnly fro = 0 o =-, nd hnc no od convrson ks plc bfor h frs cll (=. Slrly nd snc h bckwrd PCW ods h ls cll (=N) qul o ro, no rflcon wll occur h nd of h srucur. Hnc no bsorbr clls r rqurd h nd of h srucur unlk h FDFD hod. IV. RESULS AND DISCUSSION A. Coprson wh FDFD o copr h rsuls of h MM hod wh h FDFD hod, on dfc rod wh rdus r d s plcd nsd PC wvgud. Fg. 2 dpcs h powr rflcon clculd wh h FDFD (dos) nd h MM hod (sold lns). h rdus of h rods of h PCW ws kn r =0.2μ, whl h lc consn ws =0.6μ. h wvlngh n fr spc ws kn λ=.55μ. h dlcrc consn of h rods ws ssud ε =9ε 0 nd h of h surroundng du ws ε b =ε 0. h rdus of h dfc r d vrd fro 0.r o 2.0r. For h clculon of h ods h nubr of pln wvs usd ws 5 for h propgon drcon (-drcon) whl 9 sndng wvs wr usd for h rnsvrs drcon (xdrcon). h grd of h FDFD ws kn Δ=r /8 n ordr o ccoun for h sll vrons n h s of h dfc rods nd 0 PML rods wr usd long h -drcon n boh sds, ncssry n ordr o n rflcons fro h dg of h copuonl wndow [3]. No h h FDFD rqurd or hn GB of RAM n ordr o solv s sys of quons. On h ohr hnd no srous ory rqurns wr posd by h MM hod. Boh hods rqurd roughly h s oun of o produc hr rsuls wh h MM hod bng slghly fsr. As obsrvd by h fgur, hr s vry good grn bwn h wo hods nd hs vrfs h ccurcy of h MM hod. A slr grn s obnd whn h poson of h dfc rods s chngd. bl I, shows h vlus for h powr rflcon clculd wh boh hods ssung sngl dfc rod (s n Fg. 2) wh r d =r whos poson chngs ±r n hr h x or h drcon. No h h MM hod copus prcclly h s vlus for R whn h rods r dsplcd ±r long h x-drcon nd hs s no surprsng snc h srucur s syrc long hs drcon. h s s ru for h -drcon s wll. ABLE I COMPARISON BEWEEN HE FDFD AND MM MEHOD FOR VARIOUS DEFEC ROD POSIIONS Poson Powr Rflcon R MM hod FDFD hod +r (x-drcon) 0,778 0,7685 -r (x-drcon) 0,7727 0,7977 +r (-drcon) 0,8286 0,8426 -r (-drcon) 0,8286 0,8638 R,0 0,8 0,6 0,4 0,2 Mod Mchng FDFD 0,0 0,0 0,4 0,8,2,6 2,0 r d / r Fg. 2 Coprson of h MM nd h FDFD hods for sngl dfc rod nsd PC-wvgud B. Coprson wh AVM o vrfy h ccurcy of MM hod for sll gorcl prurbons of dsconnuy n PCW, w nroduc on dfc rod wh rdus r d =.2r nsd h PC wvgud nd clcul h powr rflcvy of h dvc by chngng h dfc rdus by ±% (fgur 3). h rs of h PC prrs (lc consn, rod rdus r, c) r h s s n scon IV A. h rsuls r coprd wh h AVM n fgur 4. AVM s usd o clcul h snsvy of h powr rflcvy of h PCW wh on dfc rod, wh rspc o h dfc rod rdus. o pln hs hod h Mxwll quons wr solvd n h frquncy don usng h FDFD nd h snsvy s clculd by consdrng h djon probl s oulnd n [5]. h grd of h FDFD ws kn Δ=r /6. As sn n h fgur, n xclln grn s obsrvd n rs of h slop of h vron of h Inrnonl Scholrly nd Scnfc Rsrch & Innovon 2(6)

5 World Acdy of Scnc, Engnrng nd chnology Inrnonl Journl of Elcroncs nd Councon Engnrng Inrnonl Scnc Indx, Elcroncs nd Councon Engnrng ws.org/publcon/3965 rflcvy n rs of h dfc rod rdus. hs confrs h pplcbly of h MM hod for h sudy of sll srucurl dvons n PC-bsd dvcs. Fg. 3 A PCW wh on dfc rod nroducd n h wvgud. h srucur s dvdd n wo dffrn yps of clls Fg. 4 Powr rflcvy of PCW wh on dfc rod nroducd n h wvgud. h dfc rod rdus s lrd by % nd h snsv of h powr rflcvy wh rspc o h dfc rdus s clculd wh MM nd AVM V. CONCLUSION h od chng hod hs bn ppld n h sudy of PC-bsd wvgud dsconnus. h hod s bsd n h xpnson of h fld n rs of h gnods of h clls of h srucur nd hr chng h boundry nrfcs. h hod ws vrfd by coprng o FDFD sulons. h hod hs provn o b ccur vn for sll gorcl vrons of h PC prrs. MM hod cn provd usful physcl nsgh nd cn b usful n h dsgns of PC opcl flrs bsd on wvgud dsconnus. [3] A. Mks, J. C. Chn, I. Kurlnd, S. Fn, P. R. Vllnuv nd J. D. Jonnopoulos, Hgh rnssson hrough Shrp Bnds n Phoonc Crysl Wvguds, Phys. Rv. L. 77, , 996. [4] Y. Hbno, Rcn Advncs n Hgh-Dnsy nd Lrg Scl AWG Mul/Dulplxrs Wh Hghr Indx Conrs-Bsd PLCs, IEEE J. Slcd opcs n Qun. Elc. Vol. 8, No. 6, Novbr 2002, pp [5] I. Vurgfn nd J. R. Myr Phoonc-Crysl Dsrbud-Fdbck Qunu Cscd Lsrs, IEEE J. Qunu Elcroncs, Vol. 38, No. 6, Jun 2002, pp [6] M. F. Ynk nd S. Fn, M. Soljˇcc nd J. D. Jonnopoulos Allopcl rnssor con wh bsbl swchng n phoonc crysl cross-wvgud gory, OSA Opcs Lrs Vol. 28, No. 24 Dcbr 2003, pp [7] M. Koshb, Wvlngh Dvson Mulplxng nd Dulplxng Wh Phoonc Crysl Wvgud Couplrs, Vol. 9, No. 2, Dcbr 200, pp [8]. Msuoo nd. Bb, Phoonc Crysl k-vcor Suprprs, IEEE Journl of Lghwv chnlogy, Vol. 22, No. 3, Mrch 2004, pp [9] M. Id, S. Nod A. Chunn, M. Mochuk nd. nk Chnnl Drop Flr Usng Sngl Dfc n 2-D Phoonc Crysl Slb Wvgud, IEEE J. Lghwv chnology, Vol. 20, No. 5, My 2002, pp [0] R. Cos, A. Mllon nd M. Mrnll, Bndpss Rsonn Flrs n Phoonc-Crysl Wvguds, IEEE Phoon. chn. Lrs, Vol. 5, No. 3, Mrch 2003, pp [] D. Prk, S. K, I. Prk nd H. L, Hghr Ordr Opcl Rsonn Flrs Bsd on Coupld Dfc Rsonors n Phoonc Crysls, IEEE J. Lghwv chnology Vol. 23, My 2005, No. 5 pp [2] A. fflov nd S. Hgnss Copuonl Elcrodyncs: h fn dffrnc -don hod, Arch Hous Publshrs,2000. [3] M. Koshb, Y. suj, S. Ssk Hgh-Prfornc Absorbng Boundry Condons for Phoonc Crysl Wvgud Sulons, IEEE Mcrowv nd Wrlss Coponns Lrs,Vol.,No.4,Aprl 200, pp [4] S. D. Wu nd E. N. Glyss, Fn-nubr-of-prods hologrphc grngs wh fn-wdh ncdn bs: nlyss usng h fndffrnc frquncy-don hod, J. Op. Soc. A. A, Vol. 9, No. 0, Ocobr 2002, pp [5] G. Vrons, R.W. Duon, S. Fn, Mhod for snsvy nlyss of phoonc crysl dvcs, OSA Opcs Lrs, Vol. 29, No. 9, Ocobr 2004, pp [6] R.E. Colln, Fld hory of Gudd Wvs, McGrw Hll, 992 [7] M. Skorobogy, M. Ibnscu, S. G. Johnson, O. Wsbrg,.D. Engnss, M. Soljcc, S.A. Jcobs nd Y. Fnk, Anlyss of gnrl gorc sclng prurbons n rnsng wvgud: fundnl conncon bwn polrron-od dsprson nd group-vlocy dsprson, OSA J. Opcl Soc. A. B, Vol. 9, No. 2, Dc 2002, pp [8] S. G. Johnson, P. Bnsn, M. A. Skorobogy, M. Ibnscu, E. Ldorks, J. D. Jonnopoulos, Adbc hor nd connuous coupld-od hory for ffcn pr rnsons n phoonc crysls, Phys. Rv. E 66, (2002). [9] M. L. Povnll, S. G. Johnson, E. Ldorks, J. D. Jonnopoulos, Effc of phoonc bnd gp on scrng fro wvgud dsordr, Appld Physcs Lrs, Vol. 84, No. 2, My 2004, pp [20] D. Mrcus, hory of Dlcrc Opcl Wvguds, Acdc Prss Inc, Scond Edon 997. [2] G. A. Gsll, I. R. Crc, Rcurrnc odl nlyss for ulpl wvgud dsconnus nd s pplcon o crculr srucurs, IEEE rnscons on Mcrowv hory nd chnqus, Vol. 4, No. 3, Mrch 993, pp REFERENCES [] J.D Jonnopoulos, R.D. Md nd J.N. Wnn, Phoonc Crysls, Moldng h flow of Lgh, Prncon Unvrsy Prss, 995. [2] K. Skod, Opcl Proprs of Phoonc Crysls, Sprngr-Vrlg Brln, 200. Inrnonl Scholrly nd Scnfc Rsrch & Innovon 2(6)