第三章-光纤模式理论

第三章光纤模式理论ModeTheoryforOpticalFibers主要内容阶跃折射率光纤中的场模式弱导光纤中的线偏振模光波导中模式的普遍性质波导横向非均匀性的微扰法处理纵向非均匀性与模式耦合方程直角坐标（x,y,z）柱坐标（r,,z）zyxeee,,zreee,,zyxzyxeeezrrzreee1zAyAxAzyxAzAArrrArzr11zyxzyxAAAzyxeeeAzrzrArAAzrreee12222222zyx2222211zrrrrr基矢坐标系波动光学光波导理论逻辑过程Maxwell方程边界条件波动方程场的解边界条件特征方程场的解传输常数模场分布一.阶跃折射率光纤中的场模式光纤的对称性与柱坐标系下的波动方程纵向均匀光波导中场的纵横关系Bessel方程及其解阶跃光纤中矢量模的场分布矢量模的特征方程、模式分类与命名规则矢量模的特性曲线模式的截止特性、基模与光纤的单模工作条件矢量模在光纤横截面上的场分布与光功率密度分布结构阶跃型光纤折射率剖面Stepindex2121,,nnbranarnnn1n2ab2000kj=1,2芯层，包层（r,,z）为柱坐标系0112202222222EEEEEjnkzrrrr0112202222222HHHHHjnkzrrrr波动方程(柱坐标)22,0,0002222cfcknkkkkHHEEHelmholtz把E=Er+E+Ez代入到波动方程，并在柱坐标系下展开0112202222222EEEEEjnkzrrrr柱坐标系下，横场满足的方程十分复杂，除Ez、Hz外，其它横向分量都不满足标量的亥姆霍兹方程。因而矢量解法是从解Ez、Hz的标量亥姆霍兹方程入手，再通过场的横向分量与纵向分量的关系，求其他分量。横场纵场纵横关系ttzztzttzztzjHjHjjEEeeHEee0ttzztttzztjzjzEHeHHEeE0zjrzrexp,,,EErrHHEErtrrtrrt1eeeeHeeEztzztAeAttzzAAee纵向均匀、无损、z向传输ztzzttztzzttEHnkjHEnkjeHeE222002220场分布形式传输因子对称性的波动方程光纤的圆对称性电磁场沿方向为驻波解,...2,1,0,expexpmzjjmrFEzzjrzrexp,,,EE对称性的波动方程光纤的圆对称性电磁场沿方向为驻波解,...2,1,0,expexpmzjjmrFEz0112202222222EEEEEjnkzrrrr22202222212022,nkaWnkaU222021202222nknkaWUVarFrmaWdrdFrdrFdarFrmaUdrdFrdrFd,01,01222222222122221212m阶Bessel方程m阶虚宗量Bessel方程arFrmaWdrdFrdrFdarFrmaUdrdFrdrFd,01,01222222222122221212m阶Bessel方程m阶虚宗量Bessel方程01101122222222RxmdxdRxdxRdRxmdrdRxdrRdm阶Bessel方程m阶虚宗量Bessel方程Bessel方程的解01101122222222RxmdxdRxdxRdRxmdrdRxdrRdm阶Bessel方程m阶虚宗量Bessel方程xBNxAJxRmm)(xDIxCKxRmm)(贝塞尔方程的解aUrJUJErFaUrBNaUrAJrFmmmm011aWrKWKErFaWrDIaWrCKrFmmmm022Nm(0)=Im()=芯层包层Bessel函数虚宗量Bessel函数Neumann函数虚宗量Neumann函数Ez连续：F1|r=a=F2|r=aarFrmaWdrdFrdrFdarFrmaUdrdFrdrFd,01,01222222222122221212贝塞尔函数性质J函数051015202530-0.500.51xJ0(x)J1(x)J2(x)kmkkmxkmkxJ20)2()!(!)1()(贝塞尔函数性质N函数051015202530-7-6-5-4-3-2-101xN0(x)N1(x)贝塞尔函数性质I函数00.511.522.5300.511.522.533.544.55xI0(x)I1(x)I2(x)贝塞尔函数性质K函数00.511.522.53012345678910xK0(x)K1(x)xKdxxdKxJdxxdJxKxKxKxJxJxJxJxJxmxJxJxJmmmmmmmmmmmm1010111111212121贝塞尔函数递推关系(了解，会用)kmkkmxkmkxJ20)2()!(!)1()(jmrGHrHjmrGErEmzmzexp,exp,00arWKaWrKarUJaUrJrGmmmmm,,aUrJUJErFaUrBNaUrAJrFmmmm011aWrKWKErFaWrDIaWrCKrFmmmm022电磁场的纵向分量jmUJaUrJrmUaJaUrUJHEnjEnHjEHjHEjUarHHEEmmmmrrexp,002100210000000022arjmWaKaWrKrmWaKaWrWKHEnjEnHjEHjHEjWarHHEEmmmmrrexp,002200220000000022arztzzttztzzttEHnkjHEnkjeHeE222002220电磁场的横向分量由“纵横关系式”得到返回22202222212022,nkaWnkaU212022220nknk22202nk导模条件泄漏模和辐射模横向约束横向辐射传输常数方向分量连续E|r=aH|r=aWWKaWrKnUUJaUrJnWUVjmHEWWKaWrKUUJaUrJVWUmjHEmmmmmmmm222122200222000特征方程4202221UWVkmWWKaWrKnUUJaUrJnWWKaWrKUUJaUrJmmmmmmmm光纤中电磁场模式的特征方程——由横向电场和磁场的边界条件得到00101WWKWKUUJUJ001220121WWKWKnUUJUJn不同的模式m=0WWKaWrKUUJaUrJVWUmjHEmmmm222000WWKaWrKnUUJaUrJnWUVjmHEmmmm2221222004202221UWVkmWWKaWrKnUUJaUrJnWWKaWrKUUJaUrJmmmmmmmmE0=0,Ez=0,TE模H0=0,Hz=0,TM模TE模TM模4202221UWVkmWWKaWrKnUUJaUrJnWWKaWrKUUJaUrJmmmmmmmm0,0,000HEm混合模xKxKxKxJxJxJmmmmmm11112121特征方程HE模EH模214210212212221122122212122UWVnkmWWKWKWmnnnWWKWKWmnnnUmUUJUJmmmmmm214210212212221122122212122UWVnkmWWKWKWmnnnWWKWKWmnnnUmUUJUJmmmmmmHE模：EH模：m反映了模场分布随方位角变化情况，n为特征方程根的序号WWKaWrKUUJaUrJVWUmjHEmmmm222000WWKaWrKnUUJaUrJnWUVjmHEmmmm222122200-V特性曲线22212nnaV矢量模的截止特性W=0,U=Vc,归一化截止频率截止条件特征方程归一化截止频率W001!1.15420,ln,(),02mmmWKWKWmWWKm(W)的小宗量近似：010cccVJVJV个根的第是方程nxJVnnTMTEcnn0,000,0000101WWKWKUUJUJ001220121WWKWKnUUJUJnTE模TM模特征方程W0UVc截止时的特征方程截止频率Vc不为0!!!TE0n,TM0n的截止频率4048.201,0101TMTEcV最小值TE01,TM01EHmn的截止频率20102limlimWmWWKWKWmmW01cmcmcVJVJV个根的第是方程nxJVmmnmnEHcmn0,832.31111EHcV214210212212221122122212122UWVnkmWWKWKWmnnnWWKWKWmnnnUmUUJUJmmmmmm特征方程W0UVc截止时的特征