13
Phase 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
E(x,t) E0 coskx t
kx t const. vphase / k
• 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。
vg dk / d1
Using k= n()/c0, calculate: dk /d = (n + dn/d) / c0 vg c0 /n dn/d) = (c0 /n) / (1 + /n dn/d )
Finally:
vg
v phase
/
1
n
dn
d
So the group velocity equals the phase velocity when dn/d =
0.8
D
z/L = 4
D
0.6
0.4
T1 / T0 1 z / LD 2
0.2
0.0-6
-4
-2
0
2
4
6
T/T 0
24
GVD致脉冲啁啾
T
2sgn2 z / LD 1 z / LD 2T02
T
• 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)；
E%tot (x,t) E%0 exp i(k1x 1t) E%0 exp i(k2x 2t)
Let
kave k1 k2 / 2 and k k1 k2 / 2
Similiarly, ave 1 ቤተ መጻሕፍቲ ባይዱ2 / 2 and 1 2 / 2
So: E%tot (x,t) E%0 exp i(kave x kx avet t)
Taking the real part yields the product of a rapidly varying
cosine (ave ) and a slowly varying cosine ().
16
When two light waves of different frequency interfere, they produce beats
T1 T0
1
C 2 z
T02
2
2z
T02
2
2C > 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 ：
T1min
T0 , 1 C2
C zmin 1 C2 LD
26
GVD 对超Gaussian脉冲的影响
超Gaussian脉冲
A(0,T
)
exp
Az,T
1
2
A~0,exp
i 2
2 2
i 6
3 3
iT d
Where
A~0,
A0,T
expiT
d
22
GVD 对Gaussian脉冲的影响
Consider the propagation of an initial Gaussian pulse,
A0,T
A0
exp
T2 2T02
After propagating a distance z, the pulse evolved into the following form
Note that fiber folks define GVD as the negative of ours.
Sophisticated cladding structures, i.e., index profiles have been designed and optimized to produce a waveguide dispersion that modifies the bulk material dispersion
18
Group velocity is not equal to phase velocity
if the medium is dispersive (i.e., n varies)
For
our
example,
vg
k
c0k1 c0k2 n1k1 n2k2
where k1 and k2 are the k-vectors in vacuum.
•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
i A z
i
2
A
2
2
2A T 2
A2A
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：
则NLS方程变换成如下归一化形式：
i U
sgn2 2U
2 2
N 2ez U
2U
式中N2 = LD/LNL , N是孤子阶数。
5
色散长度与非线性长度
•定义：
LD T02 / 2 , LNL 1/P0
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
LD T02 / 2 , LNL 1/P0
引入下列归一化变量
T /T0, A z, P0ez/2U z,
NLS方程变换成如下归一化形式：
i U z
sgn2 2U
2LD 2
ez LNL
U
2U
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：
z / LD
E%tot (x, t) E%0 exp(i1t) E%0 exp(i2t)
Let
ave
1
2
2
and 1 2
2
So: E%tot (x, t) E%0 exp i(avet t) E%0 exp i(avet t) E%0 exp(iavet)[exp(it) exp(it)] 2E%0 exp(iavet) cos(t)
第三章 群速度色散
文双春 唐志祥
2009年3月17日星期二
1
Contents
• 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
12
Questions
• 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 the frequency of the pulse? What is the velocity of the pulse?
0.4
0.3
2T 0
0.2
0.1
0.0
-6
-4
-2
0
2
4
6
T
21
色散致脉冲展宽的解析研究方法
Starting equation:
A z
i2
2
2A T2
3
6
3A T3
This equation can be easily solved by Fourier
transformation. The solution is
17
Group velocity
Light wave beats (continued): Etot(x,t) = 2E0 cos(kavex–avet) cos(kx–t)
This is a rapidly oscillating wave [cos(kavex–avet)] with a slowly varying amplitude [2E0 cos(kx–t)] The phase velocity comes from the rapidly varying part: v = ave / kave What about the other velocity? Define the "group velocity:" vg /k In general, we define the group velocity as: vg d /dk
If n1 n2 n,
vg
c0 n
k1 k1
k2 k2
c0 n
phase
velocity
If n1 n2 ,
vg c
19
Calculating the Group velocity
vg d /dk
Now, 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 it's easier to think of as the independent variable:
“Angular dispersion”
dn/dl
Dispersion also disperses a pulse in time: