第三章 群速度色散

20
常见脉冲形式
Intensity
1.0
Gaussian
0.9
S-Gaussian
0.8
Sech
0.7
0.5
0.6
T
FWHM
0.5
1/e
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
Az,T
1
2
A~0,exp


i 2
2 2

i 6
3 3

iT d

14
When two waves of different frequency interfere, they produce "beats"
Individual waves Sum Envelope
Intensity:
15
When two waves of different frequency interfere, they produce beats
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
• 意义：
LD： GVD开始起作用的长度； LNL：非线性效应开始起的长度
6
色散长度与非线性长度
• 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，非线性效应起主要作
用； • L >> LD、L >> LNL，色散和非线性效应
•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：
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
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)
T
• 线性频率啁啾（横过脉冲的频率变化是线性的）；
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:
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
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.
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?
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
“Angular dispersion”
dn/dl
Dispersion also disperses a pulse in time:
“Chirp”
d2n/dl2
Both of these effects play major roles in ultrafast optic11s.
Dispersion in Optical Fibers

23
GVD使Gaussian脉冲加宽
|A(z,t)|2
1.0
z/L = 0
D
z/L = 2
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
Az,T

A1
exp
T2 2T12
iz,T
A1 A0 / 1 z / LD 2 , T1 T0 1 z / LD 2 ,
z,T

sgn2 z / LD 1 z / LD 2
T2 T02

1 2
tan1
z LD
如果进一步引入归一化传播距离：
z / LD
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。
共同起作用：反常色散：Soliton; 正常色 散：Pulse Compression.
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
8
Optical dispersion
9
Negative dispersion
dn 0!
d
10
Dispersion in Optics
The dependence of the refractive index on wavelength has two effects on a pulse, one in space and the other in time. Dispersion disperses a pulse in space (angle):
So: E%tot (x,t) E%0 exp i(kave x kx avet t)
E%0 expi(kave x kx avet t)

E%0
exp
i(kave
x

avet
)
exp i(kx t) exp{i(kx t)}
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 2 / 2 and 1 2 / 2
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
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
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
来自百度文库

sgn2 2U
2LD 2
ez LNL
U
2U
4
非线性Schrodinger方程的归一化

n
dn
d

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 < vphase.
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。
2E%0 expi(kavex avet) cos(kx t)
Real part: 2E0 cos(kavex avet) cos(kx t)
17
Group velocity
Light wave beats (continued): Etot(x,t) = 2E0 cos(kavex–avet) cos(kx–t)