1、第二章 脉冲(michng)在光纤中的传输,文双春唐志祥27 July 2009,第一页，共三十六页。,你应该掌握(zhngw)什么?,光纤中的光场遵循什么规律(gul)?如何描述光场?如何描述光场与光纤介质的相互作用?如何描述脉冲？如何描述光脉冲在光纤中的传播?如何数值求解非线性Schrodinger方程?,I recommend that you grasp them.,第二页，共三十六页。,光是电磁波,Electric(E)and magnetic(B)fields are in phase.The electric field,the magnetic field,and the pr
2、opagationdirection are all perpendicular.,第三页，共三十六页。,介质(jizh)中的Maxwell 方程组,God says,let Maxwells equations govern the propagation of light!,法拉第电磁感应(dinc-gnyng)定律(随时间变化的磁场产生电场),安培定律（电流和随时间变化(binhu)的电场产生磁场）,高期定律-电荷分布产生电场,没有磁单极子,But 光与物质如何联系呢?,物质方程,Especially,对于光纤介质:=0,J=0,M=0,第四页，共三十六页。,I think Maxwel
3、l equations is too generic or too complex to solve.And how can I see the wave nature of light from these equations?,第五页，共三十六页。,Derivation of the Wave Equation from Maxwells Equations,Take of:Change the order of differentiation on the RHS:,第六页，共三十六页。,Derivation of the Wave Equation from Maxwells Equa
4、tions(contd),But:Substituting for,we have:where c2=1/00,第七页，共三十六页。,从Maxwell 方程组到关于(guny)电场的波动方程,For P=0,homogeneous Wave Equation;But here P 0,Inhomogeneous Wave Equation.The polarization is the driving term for a new solution to this equation.,Here we have used a Maxwell equation,and decomposing P=
5、PL+PNL,we obtain a general wave equation:,Further,using the relation:,矢量(shling)分析课程,感应极化描述物质效应.它与光场有什么(shn me)关系呢?,第八页，共三十六页。,感应极化(j hu)P 与光场的关系,如果瞬时性和局域性成立(chngl),则 P 与 E 的关系为,Linear Optics,Nonlinear Optics,二阶非线性,对于(duy)光纤可忽略,三阶非线性,但是,事项总是有因果性,前因后果.所以瞬时性一般不成立,那么 P 与 E 的关系如何呢?,第九页，共三十六页。,感应极化(j hu)
6、P 与光场的普适关系,如果局域性仍然成立,并考虑到三阶(sn ji)非线性,则 光纤中P 与 E 的关系为,问题(wnt)太复杂,可简化吗?,第十页，共三十六页。,光纤中 P 与E的简单(jindn)关系,极化(j hu)率是复数,实部和虚部分别与介质的折射率和吸收系数有关.,第十一页，共三十六页。,光纤中 P 与E的简单(jindn)关系,奇怪!介质的折射率怎么与光场有关(yugun)?这就是非线性光学,没什么复杂的,只不过光强改变了折射率罢了.,假设介质没有吸收(xshu),则极化率为实数,第十二页，共三十六页。,下一步(y b)做什么?,从波动方程出发利用上述关系推导出光场慢变包络(bo
7、 lu)(光脉冲)满足的非线性Schrodinger方程,光场怎么(zn me)表示?,第十三页，共三十六页。,从波动(bdng)方程到非线性Schrodinger方程,你若有兴趣,请参考试一试,考考你的数学能力.你若嫌麻烦,那么只要你承认(chngrn)这个方程并理解各项的物理意义就行了.,你在课程中已经推导了线性传输方程,用类似的方法可得到(d do)非线性Schrodinger方程,只要注意利用折射率与光场的关系就行了.,本课程的核心是非线性Schrodinger方程!如何得到它?,以后的工作全指望Schrodinger了!,第十四页，共三十六页。,非线性Schrodinger方程(fn
8、gchng),第十五页，共三十六页。,非线性Schrodinger方程(fngchng)(including higher-order terms),第十六页，共三十六页。,到底(do d)用哪个方程？需考虑哪些项？,脉冲宽度：T0 5 ps,不考虑高阶项；50fsT05ps,考虑高阶项；比较各项的相对重要性（如用色散长度、非线性长度来衡量）；特别考虑某项的作用(zuyng)时，可暂时忽略其它项的影响，如第三章、第四章。,第十七页，共三十六页。,再考察(koch)光场,光场满足波动方程(fngchng)或Maxwell方程；光场慢变包络(光脉冲)满足非线性Schrodinger方程。,光场表示
9、(biosh)：,第十八页，共三十六页。,如何(rh)描述光脉冲？,第十九页，共三十六页。,An ultrashort laser pulse has an intensity and phase vs.time.,Neglecting the spatial dependence for now,the pulse electric field is given by:,Intensity,Phase,Carrier frequency,A sharply peaked function for the intensity yields an ultrashort pulse.The pha
10、se tells us the color evolution of the pulse in time.,第二十页，共三十六页。,Temporal&Spectral Shapes of common pulses,第二十一页，共三十六页。,Second-order Phase:The Linearly Chirped Pulse,A pulse can have a frequency that varies in time.,This pulse increases its frequency linearly in time(from red to blue).In analogy to
11、 bird sounds,this pulse is called a chirped pulse.,第二十二页，共三十六页。,Chirped wave and chirped pulse,第二十三页，共三十六页。,The Instantaneous Frequencyvs.time for a Chirped Pulse,A chirped pulse has:where:The instantaneous frequency is:which is:So the frequency increases(0)or decreases(0)linearly with time.,第二十四页，共
12、三十六页。,常见(chn jin)啁啾脉冲表达式,第二十五页，共三十六页。,The Time-Bandwidth Product of a Chirped Gaussian Pulse,无啁啾(zhu ji)情况下（C=0）,Fourier-Transform Limited.有啁啾情况下（C0）,若T0不变，则谱加宽；若谱宽不变，则脉宽加宽。,第二十六页，共三十六页。,如何(rh)求解非线性Schrodinger方程,解析方法:逆散射方法(inverse scattering method)微扰法(perturbation approach，Yu.S.Kivshar,B.A.Malomed,
13、Rev.Mod.Phys.,1989,61(4):763-915)变分法(variation approach,文双春等,中国(zhn u)科学，A辑,1997,10；Anjan Biswas，J.Opt,A,2002,4:84-97)矩方法（moment method，J.Santhanam,Opt.Commun.,222:413-420）,数值方法:分步Fourier方法(split-step Fourier method)有限(yuxin)差分法(finite-difference technique)小波变换法(wavelet transform technique),第二十七页，共三
14、十六页。,数值(shz)求解NLSE,The NLSE can be generally written asThe dispersion operator,and the nonlinear operator,.For dispersion step,this equation can be easily solved by using Fourier transformation,For nonlinear step,the equation has the former solution,Thus,the solution to the NLSE is,第二十八页，共三十六页。,分步Fo
15、urier方法(fngf),第二十九页，共三十六页。,分步Fourier方法(fngf)(Split-Step Fourier Method,SSFM),第三十页，共三十六页。,分步Fourier方法算法(sun f)实现,第三十一页，共三十六页。,分步Fourier方法算法(sun f)实现,Step 1.Define the initial data(e.g.Gauss or sech);Step 2.linear propagation half a step z/2(i.e.,Fourier transform the data,multiply by the quadratic ph
16、ase factor,and invert the transform);Step 3.multiply by the nonlinear exponential term;Step 4.linear propagation a full step z(i.e.,Fourier transform the data,multiply by the quadratic phase factor,and invert the transform);Step 5.repeat step 3 until the point(L-z/2)is reached,and then branch to step 6;Step 6.linear propagation half a step z/2(i.e.,Fourier transform the data,multiply by the quadratic phase factor,and invert the transform).,第三十二页，共三十六页。,Problems about Numerical Simulations on NLS