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1、NonlinearOptics§IndicatrixMethodConsiderthedispersionrelationsforordinary&extraordinarywaves,1cOrd:w/k=k11g=kg==meno(q)eForanyangleq,no(q)=no=.eo22Ext:w/k=k22g=g(kcosq+kzsinq)1/2æcos2qsin2qöc=cç+÷ºç22÷n(q)ènoneøeezwherene=.eo11212=cosq+sinq.n(q)2n2n2eoeÞ1.Wecancontroltheindex
2、(phase)ofawavebycontrolq.®Ù2.q=90 ,ne(q)=ne,andDinzaxis.3.no£ne(q)£ne(forpositiveuniaxialcrystal).Indicatrix:Asimplegeometricmethodforfindingindexand®polarizationforwavesinagivenkdirection.x2y2z2Indicatrix:++=1n2n2n2ooe3Dellipsoidalsurface36NonlinearOpticsMoregeneral:ekxx=1oi
3、jijzne®kqnoynone(q)x®1.Givenkdirection.®2.Considertheplane^tok.®intersectsindicatrixinanellipse.®(i)minoraxisindex=nwithDinxdirection.o"Ordinarywave"(ii)majoraxisindex:37NonlinearOpticszneqnyorq22(rcosq)(rsinq)+=122none2211cosqsinq==+2222ne(q)rnone®ÙÙwithDincosqy-sinqzdirecti
4、on"Extraordinarywave"WaveSurface(kSurface)Geometricrepresentationfordispersionrelation.kzkky=ksinqqkz=kcosqkyWavesurfaceisthesurfaceoff(kx,ky,kz,w)=0.38NonlinearOpticsOrdinaryWave:2c2w=gk=(sameequationasindicatrix)k2n2o21w222=2(ky+kz)cno®Phasevelocity:inkdirectionwithamplitud
5、ewwc==kænowönoç÷ècøGroupvelocity:f(kx,ky,kz,w)=0df=0¶f¶fdki+dw=0¶ki¶w¶w¶f/¶kiÑkfvg==-=-¶ki¶f/¶w¶f/¶wÞnormaltoksurface¶f(withasaconstant)¶w39NonlinearOpticskz®nwvgo®ckqnownewcckyExtraordinarywave:2c2c2w22=cosq+sinqk2n2n2oe22k2wkzy=+222cnone®Phasevelocity:inkdirectionwithamplit
6、udewc=.kne(q)Groupvelocity:normaltoksurface®Þ(1).differentfromk.®(2).inthesamedirectionasS.Polarization(incomplexvectornotation)®®®-iwtD=Dr+iDi(amplitudewithoute)®®-iwtD(r,t)=Re{De}®®=Drcoswt+Disinwt40NonlinearOptics®®IfDr,Diindifferentdirection,®ÞD(r,t)directionwillchangeint
7、ime.¶®®®D(r,t)=-wDrsinwt+wDicoswt¶t®®Atwt=0D=Dr®®wt=p/2D=Di®Di®Dr®®•IfDandDiareparallelorantiparallelrÞlinearpolarization®®(includingeitherDrorDiiszero)®®•IfDrandDiareequalinamplitudeand^ÞcircularpolarizationOpticalRetardationPlate(l/2,l/4plate)®Ù®ÙUniaxialmedium,k^z.(Definek
8、iny)41NonlinearOpticsz(opticalaxis)®Duniaxialcrystale®D®ky®Doxd®Disa
