Electrooptic Effect The index of refraction of certain crystals can be modulated by an externally applied electric field.

Transcription

1 Electptic Effect The idex f efacti f cetai cstals ca be mdulated b a exteall applied electic field. Idex Ellipsid I geeal, the idex f efacti is detemied b the diecti f the electmagetic wave ad its plaiati. This ccept is best udestd b csideig the idex ellipsid. idex ellipsid is a 3D ftball shaped stuctue which ca be descibed mathematicall as: x x The cdiate sstem is i aligmet with the cstal axes. The cstal ma pssess the tatial smmet alg cetai axis ad the eflecti smmet agai cetai mi plaes. The cefficiets ae half legths f the ellipsid alg picipal axes. Due t smmet, sme f the cefficiets ae equal. If all f them ae equal, the ellipsid becmes a sphee. The idex is idepedet f the diecti f ppagati ad the plaiati, i.e., thee is biefigece. Mst electptic cstals ae uiaxial. I the wds, the -axis stads ut as the axis pssessig the highest degee f tatial smmet with a uique idex. e x Thee ae tw distictive idices, the dia idex ad the extadia idex. The tems have histical sigificace. B placig a piece f calcite which is biefiget tp f a page f text, thee ae tw images. While tatig the calcite cstal, e image is statia. This image is fmed b the dia a. The the image tates. It is fmed b the extadia a. F lithium ibate at 633 m, the values ae. ad.86 f the extadia idex ad dia idex, espectivel. F a electmagetic wave ppagatig alg a diecti with a agle θ fm the -axis, thee ae tw idices f the tw thgal plaiatis. The ae detemied b the half legths f tw picipal axes f the ellipse fmed b the itesecti f the idex ellipsid ad a plae which is mal t the diecti f ppagati ad passes thugh the igi. Mathematicall, the equatis t detemie them ae: cs θ si θ e Thee ae tw specific diectis f iteest, amel, θ0 ad θ90. Whe the agle is e, the beam ppagates alg the -axis. The tw thgal plaiatis ae alg the x-axis ad the -axis. Bth f them expeiece the same idex f efacti, amel,. If the agle is iet degees, e.g., the beam ppagates alg -axis, thee ae tw thgal plaiatis, alg - axix ad alg x-axis. The idices f efacti ae e ad, espectivel. Whe the cstal is subect a exteall applied electic field, the idex ellipsid becmes distted. The picipal axes ma tate as well as stetch cmpess. The mathematical epesetati f the ew idex ellipsid becmes

2 x xx x x x x Thee ae six tems ad six cefficiets. The exteall applied field ca have thee cmpets alg, x-, -, ad -axis, espectivel. T elate six cefficiets with thee quatities, we eed a 3x6 matix, i.e., thee ae 8 electptic tems i geeal. Witte i matix fm, we have xx x x E E E x Thee ae eightee tems i the electptic tes. Of cuse, t all tems ae peset because f cstal smmet. I fact, quite a few tems ae e. mg the -e tems sme have same values. Sme have same values but diffeet sig. Cstals ae guped b smmet it s-called pit gups. Each gup has a uique smbl epesetig its smmet. F example, KDP KH PO 4 belgs t the pit gup 4m. The - axis has a fu-fld tatial smmet. The x- ad -axes have tw-fld tatial smmet. The x- plae is a mi eflecti plae. If u appl a electic field alg x-axis ad get a psitive chage i the fist idex tem. Theefe, must be psitive. Hweve, b tatig the cstal 80 alg the -axis, the cstal is uchaged because f smmet. Theefe, the fist idex tem wuld chage b the same amut ad i the pevius case. Hweve, i the ew ietati, the applied field w is alg the egative x-axis. Theefe, must be egative. T eccile bth cditis, e must cclude that 0. Fllwig simila agumets, u will fid mst electptic tems e except 4, which equals 5, ad 63. Lithium ibate LiNbO 3 has bee a bechmak electptic cstal sice 976. It belgs t the pit gup 3m ad has The stgest electptic effect ca be ealied b havig the exteall applied mdulati field ad the plaiati f the beam alg the -axis. This ca be achieved with the x-cut cstal which has its suface mal t the x-axis ad a waveguide lcated betwee tw cplaa

3 electde stipes. The plaiati is alg the -axis which is paallel t the suface, theefe, it is a TE beam. It ca als be achieved with the -cut cstal. Hweve, i this case, the ptical waveguide must be ude the electde. It is TM plaied. Whe a ptical beam impiges a metallic plae, the TM plaiati ma be absbed. Theefe, a dielectic buffe lae, e.g., SiO, is eeded betwee the waveguide ad the metal electde. Based the defiiti f the electptic tes, it is staight fwad t fid b diffeetiati that 3 E The figue f meit f a electptic cstal is pptial t l t -cefficiet but als t the cube f the idex f efacti. F lithium ibate, eve with the lagest -cefficiet ad a stg mdulati field at 5 V/m, the iduced idex chage b the electptic effect is appximatel 7x0-4. It is at least e de magitude less tha the idex diffeece betwee the ce ad claddig f a tpical waveguide. T achieve a sufficietl deep mdulati, i.e., L/λπ, the legth f the electptic device is i the age f 5-0 mm. lthugh ptical waveguides ae mic i sie, the legth f a electptic device makes the aspect ati ve lage. It limits the umbe f devices which ca be itegated t e substate. Electptic effect exists t l amg igaic cstals but als i plme films. Hweve, the lg-chai mlecules must lie up i de t shw the electptic effect. This is de b plig, i.e., liig up mlecules ude a ve lage static electic field. Plig is als pefmed igaic cstals t lie up dmais f sptaeus plaiati. Plig is pefmed at a elevated tempeatue abve which the dmais ae mbile. Oce it is de, the cstal is cled dw t m tempeatue ad the dmais ae lcked it psiti. If the cstal is heated abve cetai tempeatue, the Cuie tempeatue, dmais will admie ad the electptic effect Phase mdulat The simplest electptic mdulat is a phase mdulat. It is fte used t test the pefmace f ew electptic mateials. The phase mdulat csists f l a sigle waveguide. s the i ω t kx E Ee e ptical beam tavels thugh the mdulat, the phase is chaged b the mdulati sigal. Sice the istataeus fequec f the electmagetic wave is give b dφ ω ω dt The phase mdulat ceates ew fequecies side bads i the spectum. If the phase shift is lieal pptial t time, a sigle side bad is ceated. It becmes a fequec shifte. Of cuse, the phase ca t gw with time idefiitel. B measuig the spectum usig a Fab- Pet, the pefmace f the phase mdulat ca be chaacteied. The phase mdulat ca be used as the ecdig device f digital sstems usig the phase-shift-keig methd. phase mdulat ca als pefm itesit mdulati if it is fllwed b a aale, i.e., a plaie at the utput ed f the device. The iput plaiati shuld be at 45-degee fm the picipal axis. Tw cmpets, TE ad TM, ae lauched. The expeiece diffeet phase shift

4 accdig t the -cefficiet. t the utput the plaiati state is chaged accdig t the mdulati sigal. The aale passes l e plaiati. The pwe at the utput, theefe, is mdulated. Mach-Zehde itefemetic mdulat The Mach-Zehde itefemetic mdulat csists f tw Y-baches at the iput ad at the utput. I the middle secti, thee ae tw paallel waveguide ams. Eithe e f the ams is mdulated bth f them ae mdulated b i ppsite plait s that thee is a push-pull effect. The et effect is that the electmagetic fields i tw ams have diffeet phases befe the ae cmbied i the utput Y-ucti. If the phase diffeece is eve itege multiple f π, the add up cstuctivel. The utput is. If thee phase diffeece is dd itege multiple f π, the add up destuctivel. The utput is ff. Whe the utput is ff, the pwe adiates it the substate istead f appeaig at the utput waveguide. The qualitative descipti ca be summaied as: φ E E Ee E e Ee cs I E I cs The Mach-Zehde itefemete is a e-i-e-ut device. It is a itesit mdulat but t a full fuctial switch. basic switch shuld have tw iput pts ad tw utput pts. vaiati f the Mach-Zehde itefemete is ealied b eplacig the tw Y-uctis with tw 3 db cuples. Each cuple has tw iput pts ad tw utput pts with a cuplig egi i betwee. The pts have waveguides 5-mic apat t match with fibes. Thee is iteacti betwee tw waveguides whe the ae fa apat. B usig S-shaped waveguide beds, tw waveguides ae bught tgethe t the cupig egi. I this egi, the waveguides ae quite clse. I fact, the gap is cmpaable t the dimesi f the waveguide. The expetial tail f the electmagetic field f e waveguide becmes the excitati suce f the adacet waveguide. If tw waveguides ae idetical, eeg ca flw fm e guide t the the with bth the eeg ad mmetum csevati laws satisfied. The cuples ca be electicall tued t esue a pefect splittig ati, theefe, the exticti ati which is the ivese f the -ff ati f the device ca be made high, i.e., -5 db t 30 db. φ Diectial cuple s descibed i the pevius secti, a diectial cuple is a x device. Ude the ctl f a mdulati sigal, it fuctis as a switch. Thee ae tw electmagetic fields ppagatig alg the same diecti i tw cupled waveguides. The cupled mde equatis f the amplitude f electmagetic field ae: d B d db B d is the magitude f the wave vect, i.e., π/λ ad is the cuplig cstat. These equatis ae ve simila t what we used i discussig the distibuted Bagg gatig. Hweve, hee we ae dealig with tw c-ppagatig beams. These equatis ca be mdified b itducig ew vaiables:

5 We have: Whe tw waveguides have the same, the cupled equatis ca be tasfmed it simple, secd-de dia diffeetial equatis: With a iitial cditi that the ptical beam is lauched l i waveguide, the sluti becmes: The pwe is pptial t the squae f the field, theefe, the pwe i waveguide is It is a peidic fucti with pwe cuplig betwee tw waveguides. The half peid is called e cuplig legth, i.e., the pwe is cmpleted tasfeed fm e waveguide t the the. I geeal whe values f ae diffeet, we have: - /. The tasfe matix is a uita matix with the value f its detemiat equal uit. This is the esult f pwe csevati. The tw diagal elemets ae cmplex cugate pais. If the iput pwe is lauched i waveguide, the pwe i waveguide B at the utput is The esult is alwas less tha uit uless 0. The cmplete css ve ca l happe if tw waveguides ae idetical. Othewise, a patial tasfe ccus peidicall. The flw f pwe is t detemied b which waveguide has me pwe. It is detemied pimail b the phase. Tw waveguides ca be csideed as e geeal waveguide i the eige mde appach. Thee ae tw supemdes, smmetic ad asmmetic. The add tgethe i phase t pduce the cditi that all pwe is i e waveguide. If added ff phase, the esult is pwe i the ppsite waveguide. Thee ae tw supemdes ppagate with its w phase velcit. The ae i phase at e lcati ad ut f phase at athe. peidic pwe tasfe is ealied. The pictue f supemde is much me cveiet i descibig a multi-waveguide sstem. Be B e e d db B e d d d d cs cs I I π l c B B si cs si si si cs si B

6 The switchig diagam f a diecti cuple is peseted i the L-L cdiate sstem. The css state epesets islated pits alg the -axis. O the the had, the ba state, i.e., all eeg eds i the igial waveguide, epesets a seies f quate-cicle cuves. T ealie a pefect css state wuld equie stiget fabicati ctl. This is ve difficult t achieve. I de t vecme the pblem, the ccept f alteatig electde was itduced. T T T T T T T T T T T T T T T T T diectial cuple with tw electde sectis ca be csideed as cascaded cuple. The T T 0 T T veall tasfe fucti is the pduct f idividual tasfe matices. Whe the tw sectis ae mdulated b the same vltage but ppsite plait, the tw tasfe matices ae elated b cmplex cugate. F all iput pwe i e waveguide, we have: The E-field i the fist guide at the utput is T si The secd tem has plet pptuities t becme uit, i.e., the E-field i the fist waveguide is e a cmplete cssve state. I the switchig diagam, the css state becmes a seies f cuves ust like the ba state. fte a device is made with a fixed L, cmplete switchig ca alwas be ealied b chagig the mdulati vltage. exticti ati f bette tha -30 db ca be ealied. Such a device has sufficiet pefmace i fmig ptical switchig fabics. Seveal vel device ccepts wee develped fllwig the alteatig switch. T educe side lbes i the switchig chaacteistic, the cuplig cstat ca be tapeed. The Fuie tasfm f a squae fucti is a sic fucti with a seies f side lbes. B smthig the squae fucti it a Gaussia fucti, the tasfmed sic fucti becmes als a Gaussia fucti. ll side lbes ae cmpletel suppessed. fte side lbes ae suppessed, e ca desig a plaiati idepedet switch b fist equaliig the cuplig cstat f tw plaiatis ad b mdulate the device with a lage eugh vltage. Bth the stg electptic tem ad the weake electptic tem ca be cmpletel switched t eithe the ba state the css state.