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Introduction to Waveguide Optics 2572-4 Winter College on Optics: Fundamentals of Photonics - Theory, Devices and Applications 10 - 21 February 2014 Introduction to Waveguide Optics Jiri Ctroky Institute of Photonics and Electronics AS CR, v.v.i., Prague Czech Republic IntroductiontoWaveguideOptics JiƎítyroký [email protected] Instituteof Photonics and Electronics AS CR, v.v.i., Prague,CzechRepublic WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy WhereIcomefrom… InstituteofPhotonics andElectronicsASCR,v.v.i. AcademyofSciencesoftheCzechRepublicisanonͲuniversityinstitutionforbasicandappliedresearchconsistingof54independentinstitutes WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy 1 Examplesofwaveguidestructures opticalfibre channeloptical “ARROW” waveguide (antiresonantreflectingOW) longitudinally uniform microring angularly plasmonicwaveguide resonator uniform metal subwavelengthgrating photoniccrystal waveguide waveguide “gain/loss”waveguide loss gain longitudinally periodic etc… WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy Theoreticalfundamentalsofopticalwaveguides Tentativelistoftopics • Planarwaveguides;waveguidemodes,theirproperties.Guidedandleakymodes. Othertypesofwaveguiding,guidingbyasingleinterface • Waveguidebends,whisperingͲgallerymodes,circularresonators • Channelwaveguides,approximateanalyticalmethods. • Morecomplexwaveguidestructures.FundamentalsofarigorouscoupledͲmodetheory • Introductiontomodalmethods;transfermatrixmethod,basicsofthefilmmodematching • Periodicmedia,Blochmodes,originofthebandgap,SWGwaveguides,(photoniccrystals) • “Canonical”waveguidestructures:YͲjunctions,directionalcoupler,twoͲ andmultimode interferencecouplers,microresonators • Plasmonicwaveguidesandstructures.Surfaceplasmonsensing.HybriddielectricͲplasmonic slotwaveguide,plasmonicdevices • Waveguidestructureswithlossandgain;asymmetricgratingcouplers,(“PTͲsymmetric” waveguidestructures) • … Basicrequirements:Theoryofelectromagneticfield,Maxwellequations WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy 2 Basicmath&physbackground Dielectric(possiblyalsometallic)nonͲmagneticlinearsourceͲfreemedium, timeͲharmonicdependenceofelectromagneticfield: (,)rErtit Re ()exp(Z ) E ^ ` uEH iZP0 ,0 D (,)rHrtit Re^` ()exp(Z ) 2 H uHEB inZH0 ,0 22 2S k ZPH DE H n2 000O 0 22 BH P0 kk 00H kn 1 kn (phase velocity) PlaneͲwavesolution: vcZ iikr kccc r k r EE 00eee E 1 dk ng (group velocity) iikr kccc r k r vdg Z c HH 00eee E dn()Z dn 2 nn Z group index kk cccini k, H cccH g ddZZ cccc222 dn kk k0 H complex nnO , typically larger than 2 wavevector dO 2kkccc k0 H cc WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy Fieldattheinterfacebetweentwomedia Planewaveincidentonaplanarinterface iiiikrititiikrr kr kr krr kr EEii 000e, EE rr e, EE tt e, HH i i 0 e, H r H r 0 e, HH t t 0 e TEpolarization x k x E k E t TMpolarization t t EHH,, t n n yxz 2 H yxz,,EE 2 Ht H z z t nn E n 12 E r 1 k r n ki R E i R H 1 i H k i i r k Ei r r r Hi Hi 00 00222222 kxzkxzir,01 kN rJJJJ ir ,, t kNNnNn 02 t, 1 ir ,1 , t t 2 Fieldcontinuityconditionsat x 0: 0000 0 0 izkziiizkzrr iz kz iz kz iz kz r iz kz r Eeir000 Ee Ee t, He i 0 H r 0 e He t 0 NNNNn sinT 22 22 irt1 i JJ11 nN, 2 nN 2 WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy 3 Fresnelcoefficients 22 TE Er J12 J TM Hnnr JJ11 2 2 R R 22 Ei JJ12 Hi JJ11nn 2 2 22 22 nN12 nN 22 2 22 2 nn21 N nn 1 2 N 22 22 nN12 nN 22 2 22 2 nn21 N nn 1 2 N criticalangle ) | TM TM totalreflection R ( R | , | , arg TE ) R | TE R arg( “regionofinterest” ofwaveguideoptics Brewster angle Angle of incidence (deg) Angle of incidence (deg) TM TM i ) TE Re ; TE i ) Re ; 2 ªº22 22 TM §·nNn12 N n ) 2arctan«»¨¸ . )TE 2arctan2 . «»n nN22 22 ¬¼©¹2 1 nN1 WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy Thesimplestwaveguide:two(perfect)conductors x R kxzr kN(),rJ 00 d 0 k E k RTE yref, 1 0 z E y R yinc, perfectconductors Twoequivalentphysicalpictures: TM H yref, R 1 1.aseriesofsuccessivereflectionsofaplanewave H yinc, 2.twoplanewavespropagatingupwardsanddownwards Inbothcases,the“nonzerowave”inthewaveguidecanexistonlyunder the“conditionoftransverseresonance” (forTMonly) 2 Rikdexp(200JJS ) 1, or 2 kdmm 2 , 0, 1, 2, ! Wavescanthuspropagateonlyasdiscretewaveguidemodes withpropagationconstants EETE TM kN22 k 2 k 22 J k 2 m S d2 …polarizationdegeneracy mm00 m m (exceptform =0) for md! 2O …evanescentmodes ExzEym(,) 0 sin mxdSE exp( i z ) Thetransversefielddistributionshavetheforms H ym(,)xz H0 cos mSE xd exp( i z ) WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy 4 Dielectric slab waveguide x nnncswdd Total reflection at both interfaces: nNnwcs!!, 22 22 nc JJw nN w, cs,, iNn cs R c d Condition of transverse resonance: nw z 0 R RRexp(2 ikJ d ) 1, or ns s cs0 w 11 )()Nkd J arg R arg R S m tot0 w22 s c ªºªºQQ22 22 22 §·nNnwms §· nNn wmC ­ 0 (TE) kd0 nwm N arctan«»«»¨¸ arctan ¨¸ mS , Q ® «»«»nnnN22 nN 22 2 (TM) ¬¼¬¼©¹scwm ©¹ wm ¯ TE TM Dispersion equation for Nm shows polarization birefringence, NNmm! WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy Effectivethickness&“periodofpropagation” x Total internal reflection is linked up ddR) 1(arg) LGHa LGH with the Goos-Hänchen shift, dkdNE 0 k0 J a ªºNn22 TE 22dN«»sc, d LGHs,c arctan , kdN«»22 22 2 2 z d 0 nNw kNn() nN 0 eff ¬¼0,csw L k J ªº2 Nn22 GHs 0 s TM 2 d «»nw sc, LGHs,c arctan 222 kdN0 «»nnN Lp ¬¼sc, w 2N nn22() n 2 n 2 wsc,, w sc . 22 2 2nn42 N 2 nNn 422 kNn0, sc nN w s,,cw w sc TE ddkeff 0 J s J c (and a similar, somewhat more complicated expression for TM) – effective thickness d)tot 2 dk 0J w d N Lp LLGSs GSc 2 d LL GSs GSc … “period of propagation” dkdNE 22 0 nNw These “ray-optic concepts” are useful also for inherently “wave-optic” phenomenon of optical waveguiding. WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy 5 Dispersion diagram (examples) Asymmetricwaveguide 1,60 TE0 nn 11.51.6 n TM cs w 1,58 0 All modes exhibit cut-off 1 1,56 2 3 eff 4 N 1,54 5 6 1.60 7 1,52 TE 8 0 TM 9 1.58 0 10 1 1,50 2 0246810 eff 1.56 3 N 4 d/O 1.54 5 6 7 Symmetric waveguide 1.52 8 9 nncs15.. n w 16 10 1.50 Number of TE and TM modes identical, 0246810 TE and TM always exist d/O 0 0 WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy Distributionofmodesofaslabwaveguide TE guiding layer 0.2 0 TE 1 TE 2 0.1 TE 3 TE 4 0.0 substrate cover Mode field amplitude -0.1 n = 1,5 n = 1,0 s c -0.2 n = 1,6 w -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 x coordinate (μm) WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy 6 Electromagnetic theory of a slab waveguide w Planarwaveguideasastructurewith1Dpermittivtydistribution:H ();x { 0 wy 1.TEpolarization: EHHyxz,, 2.TMpolarization: H yxz,,EE izE Eigenmodefield: ExzExyy(,) ()e , etc. i dE() x i dHy () x Hx() y , Ex() , z z 2 dx ZP0 dx ZH0nx() E E Hxxy() Ex (), Exxy() 2 Hx (), ZP0 ZH0nx() dH() x dE() x z iHEZH() x i n2 () xEx () z iEEZP() x i H () x dx xy0 dx xy0 ­½d 2 ­½ddªº1 kn22 x E x E 2 E x °°2222 ®¾2 0 yy ®¾nx «»2 knxHx0 ()yy E Hx () ¯¿dx ¯¿°°dx¬¼«»nx dx 2 EigenvalueequationsforeigenfunctionsExyy() or Hx()andeigenvaluesE WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy Analogybetweenaplanarwaveguide andapotentialwellinquantummechanics FieldequationforaTEmode Schrödingerequationforaparticle inaplanarwaveguide inapotentialwell 2 22 1()dEy 22 = dx\ 22nxE()yy NE 2 Vx()\\ () x W () x kdx0 2m dx Ex() \ ()x y Thereisnotsuchanexactanalogy 2m forTMpolarization,butitsbehaviour k isverysimilar nx2 () 0 = 2 2 nx() Vx() Vx() N0 guidedmodes 2 N 2 W x N1 substratemode freeparticle reflectionfrom radiationmode thepotentialbarrier W x 1 boundstates W0 WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy 7 Guidingofopticalradiationinadielectric waveguide x guided substrate superstrate mode mode mode nc nc nx() 0 nw nw guidedmode ns ns “substrate”mode Guidedmodes: kn00csd knE kn 0 g radiationmode Radiation (substrate) modes: kn000csgE kn kn Radiationmodes(superstrate): E dkn000csg kn kn WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy Orthogonality of eigenmodes Itcanbeshownthatfieldsofguidedmodes (fromthediscretespectrum) areorthogonal, 1 f E EHz()x u ()xdx0 m GE , kN. 2 ³ mnE mnmm0 f m Forradiationandevanescentmodes,theorthogonalityconditionsounds f 1 E (inthesense EH(,xxdxEE )u (,cc ) z0 G (EE ) 2 ³ n E oftheprincipalvalue f oftheintegral) Radiationandevanescentmodes arealwaysorthogonal todiscreteguidedmodes: 1 f EHz(,xxdxE )u ()0 0, 2 ³ n f Forlosslesswaveguides,
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