第三章 群速度色散

1、文双春文双春唐志祥唐志祥2009年年3月月17日星期二日星期二1 非线性非线性Schrodinger方程的归一化方程的归一化 色散致脉冲展宽色散致脉冲展宽 GVD对啁啾脉冲的影响对啁啾脉冲的影响 高阶色散效应高阶色散效应 GVD对光通信系统的限制对光通信系统的限制2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：AAT

2、AAizAi222222/200/, ,zT TA zP eU zUULeULzUiNLzD22222sgn0220/1 ,/PLTLNLD4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。UUeNUUiz222222sgnDLz/5定义：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：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 ,/PLTLNLD6 L LD、L LNL，色散和非线性效应都不重要； L LD、L LNL，GVD起主要作用； L LD、L LNL，色散和非线性

3、效应共同起作用：反常色散：Soliton; 正常色散：Pulse Compression.7 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8Optical dispersion9Negative dispersiondnd 0!10Dispersion in OpticsThe dependence of the refractive index on wavelength has two effects on a pulse, one in space and the other in time. Disp

4、ersion disperses a pulse in space (angle):Dispersion also disperses a pulse in time:“Chirp”d2n/dl2 “Angular dispersion”dn/dlBoth of these effects play major roles in ultrafast optics.11Sophisticated cladding structures, i.e., index profiles have been designed and optimized to produce a waveguide dis

5、persion that modifies the bulk material dispersionNote that fiber folks define GVD as the negative of ours.Dispersion in Optical Fibers12Questions Does dispersion affect the propagation of a plane wave? A wave packet (pulse) comprises of a number of plane waves with different frequencies. What is th

6、e frequency of the pulse? What is the velocity of the pulse?13Phase velocity and Group velocity 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),(014When two waves of different frequency interfere, they produce beatsIndividual wavesSumEnvelopeIntensi

7、ty:15When two waves of different frequency interfere, they produce beats01021212000( , )exp()exp()Let and 22So: ( , )exp ()exp () exp()exp()exp() totavetotaveaveaveEx tEitEitEx tEittEittEititit 0 2exp()cos().aveaveEittTaking the real part yields the product of a rapidly varyingcosine () and a slowly

8、 varying cosine (16When two light waves of different frequency interfere, they produce beats011022121212120( , )exp ()exp ()Let /2 and /2Similiarly, /2 and /2So: ( , )exp () totaveavetotaveaveEx tEi k xtEi k xtkkkkkkEx tEi kxkxtt 0000 exp ()exp () exp ()exp () 2exp ()cos()Real part: 2cos(aveaveaveav

9、eaveaveaveavEi kxkxttikxtEi kxtikxtEi kxtkxtEkx )cos()etkxt17Group velocityvg d /dkLight wave beats (continued):Etot(x,t) = 2E0 cos(kavexavet) cos(kxt)This is a rapidly oscillating wave cos(kavexavet) with a slowly varying amplitude 2E0 cos(kxt)The phase velocity comes from the rapidly varying part:

10、 v = ave / kaveWhat about the other velocity?Define the group velocity: vg /k In general, we define the group velocity as:18Group velocity is not equal to phase velocityif the medium is dispersive (i.e., n varies)0 1021 1221200121212For our example, v where and are the k-vectors in vacuum.If , vphas

11、e velocity ggkc kc kn kn kkkcckknnnn kkn12 If , vgnnc19vg d /dkNow, is the same in or out of the medium, but k = k0 n, where k0 is the k-vector in vacuum, and n is what depends on the medium.So its easier to think of as the independent variable:Using k= n()/c0, calculate: dk /d = (n + dn/d) / c0 vgc

12、0/n dn/d) = (c0 /n) / (1 + /n dn/d )Finally:So the group velocity equals the phase velocity when dn/d = 0,such as in vacuum. Otherwise, since n increases with , dn/d 0, and: vg 0, 上或正啁啾(脉冲前沿红移，后沿蓝移)；在反常色散区， V蓝；而在反常色散区， V红 0时，脉冲单调展宽； 2C 0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, 0TTiCATA22022202011T

13、zTzCTTDLCCzCTT2min20min11 ,126GVD 对超对超Gaussian脉冲的影响脉冲的影响 超Gaussian脉冲mTTiCTA2021exp), 0(-8-40480.00.20.40.60.81.0m = 3C = 0 IntensityT/T0 z/LD = 0 z/LD = 1 z/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

14、定性结果类似于Gaussian脉冲。27三阶色散效应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T0 1ps的超短脉冲。 相对重要性用三阶色散长度 ，来衡量。当L D 0时，振荡出现在脉冲后沿； 3 0时，振荡出现在脉冲前沿； 2 = 0时，振荡幅度增大，谷底逐渐趋于零。-10-505100.00.20.40.60.81.0z = 5LD IntensityT/T0 z = 0 2 = 0 LD = LD-20-15-10-5051015200.00.20.40.60.81.01.22 = 0, m = 312345z/LDIntensit

15、yT/T029一般条件下，线性色散介质中啁啾高斯脉冲展宽的表达式为若光源线宽远小于脉冲带宽，则光源线宽对脉冲展宽的影响可以忽略，即Vw = 0。dzUzUNidTTzUTNTdTTzUdTTUNTTnncnncnc,1 , 0 ,22222spectrum. sourceGaussian theof width rms theis ,24211212102303222220222202202VzVCzVzC30由RMS脉宽表达式可得如下一般性结论：1.二阶和三阶色散对高斯脉冲展宽均有影响；2.二阶色散的作用依赖于2C的符号，而三阶色散的作用并不依赖于3和C的符号；3.在严格零色散波长处传输的啁

16、啾脉冲没有脉冲压缩过程。然而，即使偏离零色散波长很小的量，也可能导致脉冲的压缩；4.对任意脉冲形状，当传输距离较长时，RMS脉宽与传输距离成线性关系。31233223222201012,! 2 6 Assume that ( , ) is normalized, i.e., 1, we have, where all the coefficients only relates to initial pulse,nnn-iUiiHU HznTTTU z TUdTTaa zTbb zb z 0*000100*2*2*2000100200 ( )(0, ),and are defined as, , where ,1, , ,2-U TUTaU TU dTai UH T U dTH THTTHbU T U dTbi UH TU dTbUHH TU dT 32物理意义：决定脉冲形状的不对称性；：脉冲展宽的量度；33自学1. Agrawal, Sec. 3.4 色散管理；2. 光纤通信技术，Sec. 3.4 比