Waves in dielectric media. Waveguiding: χ (r ) Wave equation in linear non-dispersive homogenous and isotropic media

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1 Wves i dieletri medi d wveguides Setio 5. I this leture, we will osider the properties of wves whose propgtio is govered by both the diffrtio d ofiemet proesses. The wveguides re result of the ble betwee the the diffrtio d ofiemet. The oept of wve propgtio s re-emissio proess. Wves i dieletri medi P ε χ r, t, θ, Wveguidig: χ r Dispersive: χ t or χ ν Nolier: χ P t t χ t t' t' dt' Wve equtio i lier o-dispersive homogeous d isotropi medi No-dispersive medi-- here it mes istteous. The vetor of the polriztio desity is istteously followig tht of the eletri field. P ε χ, Where χ is the eletri suseptibility whih does ot depeds o r, t or. From the reltio D ε P betwee the eletri displemet d polriztio, d D ε The dieletri ostt is give by ε ε χ Mwell s equtios re D H ε H µ H

2 Whih led to the sme form of wve equtio s i free spe eept tht the speed of light is redefied. U U Where χ ε Net, we will wor out more geerl pproh to llow P to deped o. Wves i Nolier Medium Wht is olier medium? P is futio of With the olier depedey, P ot be ftored ito. D H H D P µ µ µ 3 ε µ It follows tht P µ 7 where P is olier futio of. P be t differet frequey from tht of. This equtio will be used etesively lter. The polriztio vetor beomes the soure whih plys mjor role i olier frequey geertio, lser, d wveguide ouplig to be disussed lter. mples. The P my oti multiple powers of. As result, ew frequeies my be geerted. Moohromti eletromgeti field i medium Whe both d H re hrmoi wves of frequey, Mwell s equtios for odispersive medium beome B jb D jd B Where D ε P d µ H. These reltios will led to the Helmholtz equtio B U U 8

3 where the wve umber i the medium ε Whe the ide of refrtio is futio of positio, the U r U 9 Guided modes i symmetri oe-dimesiol wveguide To give physil pitures of wveguide d wveguide modes usig the oepts of wves bouig betwee two prllel iterfes: Oe-dimesiol slb wveguide X X/ Z Y X-/ For wve propgtig i the z-diretio, the solutio of the Helmholtz equtio 8 my be simplified by ssumig tht the eletri field is hrmoi wve give by r e j z The Helmholtz equtio be seprtely listed for eh regio. 3

4 4 Regio : Regio Regio 3 The field t the boudry must stisfy the boudry oditio t the iterfes., mely the tgetil ompoet of d H to be otiuous ross the boudry. Cosider the T mode eletri field prllel to the ple, symmetri solutio of the followig form / / os / / for p C d for C e y q y < < < > Where p d h stisfies the followig reltio p q 3 I order to hve guided mode, both p d q must be rel umbers so tht the wves do ot propgtes i the diretio, the modl propgtig ostt stisfies the followig reltio: > > 4 The boudry oditios led to the followig equtio for the eige vlue. t 5 Sigle mode Multimode

5 The symmetri wveguide lwys hs solutio. Depedig o the steps of the refrtive idies, there be more th oe mode. From the figure for the two-mode situtio, < π π or < λ < 6 The ftor is lso the umeril perture of the wveguide. The umeril perture of wveguide N.A.si θ θ θ Core Clddig 5

6 6 The umber of modes tht be supported by wveguide is λ π A N M.. For sili sigle mode fiber of ore dimeter 5 miros d NA.,. Fid the ide step. p θ

7 Bem profiles From q. 3 p or p Iterl field distributio The bem profiles of the guided modes be uderstood s beig the iterferee friges of the two wves bouig betwee the two iterfes. The lowest order mode hs the smllest p vlue d o ode. The higher order mode hs lrger p vlues d lrge umber of odes. Field i the lddig The field i the lddig derese epoetilly with iresig diste from the ore. The lrger q vlue, the fster the dey. The propgtio ostt is boud withi the rge > >. The lowest order mode hs the lrgest vlue for q. The eletri field is more ofied i the ore. The higher-order mode hve smller vlues for q d the eletri field eteds more ito the lddig. This my be outer ituitio why? Group veloity of guided modes The group veloity, the veloity of eergy or wve pet tht is detetble, is give by Group veloity v g From q. 3, p We epress i terms of d p i terms of tθ. seθ Where θ is the oblique gle betwee p d. The lowest order mode hs the smllest θ. The group veloity is give by v g seθ 7

8 Thus the group veloity of the higher-order modes with lrger θ is lrger. This outer ituitio. Why? / Group veloity / Lowest-order mode Highest-order mode θ Wveguide with qudrti refrtive ide distributio Returig to q. 8 U U 8 where the wve umber i the medium ε Whe the ide of refrtio is futio of positio, the U r U 9 ε r U U The solutios re the eige modes of the wveguide. If the wve propgtes i the z-diretio iz U u e 8

9 9 ε ε 4 [ is os wher e e u Try u d u d For u u T Cse I rel umber The eletri field is z j e e Wvefrot is plr with Gussi distributio. Bem wist t /e is proportiol to the fourth root of. Cse II pure imgiry umber j i gi guide It is ustomry to epress the sptil vritio i terms of the rel d imgiry prt of the refrtive ide. j z j i i i e e The wveguide supported by the imgiry prt of the refrtive ide is ow s the giguide. The power distributio is lso Gussi. The wvefrot is urved surfe govered by. t t os z i i

10 The wvefrot is ylidril surfe. Problems:. For optil fiber of ore dimeter 5 miros, estimte the differee i ide refrtio betwee the ore d the lddig to support sigle mode for λ-µm. How my modes this wveguide support for λ.5 µm.?..prove tht the umeril perture, N.A, of wveguide is give by. 3. Fid the mimum disprity i group veloity of the differet modes i wveguide. Assumig tht the lowest possible oblique gle is d the mimum gle is determied by the totl iterl refletio betwee the ore d lddig. 4. Two wves of wvelegth λ d propgtig i d diretios overlp i spe. The iterferee betwee the wves results i sttiory friges. Fid the frige spig d the diretio of the frige lies i reltio d. Seth the frige lies. Disuss the limitig ses whe d -.. Try to use the reltio for frige spig to epli the frige spig of the double slit Youg s eperimet.

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