第三章-群速度色散

文双春唐志祥2009年3月17日星期二第三章群速度色散1Contents•非线性Schrodinger方程的归一化•色散致脉冲展宽•GVD对啁啾脉冲的影响•高阶色散效应•GVD对光通信系统的限制2非线性Schrodinger方程的归一化•为什么归一化？简洁便于比较相对重要性标准•如何归一化？每一个量分别选取一把参考尺子去度量。一般来说，脉冲宽度用初始脉宽去度量；传播距离用色散长度去度量；脉冲振幅用初始功率的平方根去度量。3非线性Schrodinger方程的归一化设入射脉冲的初始宽度和峰值功率分别为T0和P0，定义色散长度LD和非线性长度LNL：引入下列归一化变量NLS方程变换成如下归一化形式：AATAAizAi222222/200/,,,zTTAzPeUzUULeULzUiNLzD22222sgn0220/1,/PLTLNLD4非线性Schrodinger方程的归一化如果进一步引入归一化传播距离：则NLS方程变换成如下归一化形式：式中N2=LD/LNL,N是孤子阶数。UUeNUUiz222222sgnDLz/5色散长度与非线性长度•定义：其中T0和P0分别为入射脉冲的初始宽度和峰值功率。•意义：LD：GVD开始起作用的长度；LNL：非线性效应开始起的长度0220/1,/PLTLNLD6色散长度与非线性长度•LLD、LLNL，色散和非线性效应都不重要；•LLD、LLNL，GVD起主要作用；•LLD、LLNL，非线性效应起主要作用；•LLD、LLNL，色散和非线性效应共同起作用：反常色散：Soliton;正常色散：PulseCompression.7色散致脉冲展宽•色散•相速度和群速度•常见脉冲形式•色散致脉冲展宽的解析研究方法•色散致高斯脉冲展宽•色散致双曲正割脉冲展宽•色散致超高斯脉冲展宽8Opticaldispersion9Negativedispersiondnd0!10DispersioninOpticsThedependenceoftherefractiveindexonwavelengthhastwoeffectsonapulse,oneinspaceandtheotherintime.Dispersiondispersesapulseinspace(angle):Dispersionalsodispersesapulseintime:“Chirp”d2n/dl2“Angulardispersion”dn/dlBothoftheseeffectsplaymajorrolesinultrafastoptics.11Sophisticatedcladdingstructures,i.e.,indexprofileshavebeendesignedandoptimizedtoproduceawaveguidedispersionthatmodifiesthebulkmaterialdispersionNotethatfiberfolksdefineGVDasthenegativeofours.DispersioninOpticalFibers12Questions•Doesdispersionaffectthepropagationofaplanewave?•Awavepacket(pulse)comprisesofanumberofplanewaveswithdifferentfrequencies.Whatisthefrequencyofthepulse?Whatisthevelocityofthepulse?13PhasevelocityandGroupvelocity•相速定义为与行波场保持固定相位的观察者前进的速度。沿光纤轴传播的单色波可以描述为•群速定义为与群传播包络保持固定相位的观察者前进的速度。kvtkxtkxEtxEphase/const.cos),(014Whentwowavesofdifferentfrequencyinterfere,theyproducebeatsIndividualwavesSumEnvelopeIntensity:15Whentwowavesofdifferentfrequencyinterfere,theyproducebeats01021212000(,)exp()exp()Letand22So:(,)exp()exp()exp()[exp()exp()]totavetotaveaveaveExtEitEitExtEittEittEititit02exp()cos()).aveaveEittTakingtherealpartyieldstheproductofarapidlyvaryingcosine()andaslowlyvaryingcosine(16Whentwolightwavesofdifferentfrequencyinterfere,theyproducebeats011022121212120(,)exp()exp()Let/2and/2Similiarly,/2and/2So:(,)exp()totaveavetotaveaveExtEikxtEikxtkkkkkkExtEikxkxtt0000exp()exp()exp()exp{()}2exp()cos()Realpart:2cos(aveaveaveaveaveaveaveavEikxkxttikxtEikxtikxtEikxtkxtEkx)cos()etkxt17Groupvelocityvgd/dkLightwavebeats(continued):Etot(x,t)=2E0cos(kavex–avet)cos(kx–t)Thisisarapidlyoscillatingwave[cos(kavex–avet)]withaslowlyvaryingamplitude[2E0cos(kx–t)]Thephasevelocitycomesfromtherapidlyvaryingpart:v=ave/kaveWhatabouttheothervelocity?Definethegroupvelocity:vg/kIngeneral,wedefinethegroupvelocityas:18Groupvelocityisnotequaltophasevelocityifthemediumisdispersive(i.e.,nvaries)010211221200121212Forourexample,vwhereandarethek-vectorsinvacuum.If,vphasevelocityggkckcknknkkkcckknnnnkkn12If,vgnnc19vgd/dkNow,isthesameinoroutofthemedium,butk=k0n,wherek0isthek-vectorinvacuum,andniswhatdependsonthemedium.Soit'seasiertothinkofastheindependentvariable:Usingk=n()/c0,calculate:dk/d=(n+dn/d)/c0vgc0/ndn/d)=(c0/n)/(1+/ndn/d)Finally:Sothegroupvelocityequalsthephasevelocitywhendn/d=0,suchasinvacuum.Otherwise,sincenincreaseswith,dn/d0,and:vgvphase.CalculatingtheGroupvelocity1v/gdkdvv/1gphasednnd20常见脉冲形式-6-4-202460.00.10.20.30.40.50.60.70.80.91.01/e0.52T0TFWHMIntensityTGaussianS-GaussianSech21色散致脉冲展宽的解析研究方法Startingequation:ThisequationcanbeeasilysolvedbyFouriertransformation.ThesolutionisWhere33322262TATAizAdTiiiATzA332262exp,0~21,dTiTAAexp,0,0~22GVD对Gaussian脉冲的影响ConsiderthepropagationofaninitialGaussianpulse,Afterpropagatingadistancez,thepulseevolvedintothefollowingform20202exp,0TTATADDDDDLzTTLzLzTzLzTTLzAATziTTATzA1202222012012121tan21/1/sgn,,/1,/1/,2exp,23GVD使Gaussian脉冲加宽-6-4-202460.00.20.40.60.81.0|A(z,t)|2T/T0z/LD=0z/LD=2z/LD=4201/1/DLzTT24GVD致脉冲啁啾•线性频率啁啾（横过脉冲的频率变化是线性的）；•在正常色散区，0,上或正啁啾(脉冲前沿红移，后沿蓝移)；在反常色散区，0,下或负啁啾(脉冲前沿蓝移，后沿蓝红移)；•在正常色散区，V红V蓝；而在反常色散区，V红V蓝。仅当所得的频谱分量同时到达时，脉冲宽度才能保持不变。不同频谱分量在传输过程中的任何延迟都将导致脉冲展宽。TTLzLzTDD2022/1/sgn225GVD对啁啾Gaussian脉冲的影响啁啾Gaussian脉冲传输z距离后其脉冲宽度变为2C0时，脉冲单调展宽；2C0时，脉冲先压缩再展宽，最小脉宽和达到最小脉宽的光纤长度表达式：202021exp,0TTiCATA22022202011TzTzCTTDLCCzCTT2min20min11,126GVD对超Gaussian脉冲的影响超Gaussian脉冲mTTiCTA2021exp),0(-8-40480.00.20.40.60.81.0m=3C=0IntensityT/T0z/LD=0z/LD=1z/LD=2•超Gaussian脉冲比高期脉冲展宽得快，且形状也发生畸变；•超Gaussian脉冲有较锐的前后沿，因而有较宽的谱宽。较宽的频谱导致较快的脉冲展宽率。•超Gaussian脉冲的阶数越大，脉冲展宽的越快；•当超Gaussian脉冲有初始啁啾的情况下，脉冲展宽程度依赖于2C的符号，其定性结果类似于Gaussian脉冲。27三阶色散效应两种情况下需考虑三阶色散效应：•(1)零色散波长附近，即20时；•(2)脉冲宽度T01ps的超短脉冲。•相对重要性用三阶色散长度，来衡量。当L′D=LD时，三阶色散效应起主要作用。303/DLT28三阶色散效应对脉冲传输的影响•TOD引起脉冲形状畸变，在其一个沿附近形成非对称的