第三章 光纤系统中的偏振效应
1
内容提要

背景介绍

偏振光的表述



琼斯矢量、斯托克斯矢量、帮加球
琼斯矩阵、缪勒矩阵
偏振光在双折射晶体中的传播

光纤系统中的主偏振态(PSP)

光纤系统中的偏振模色散(PMD)
2
参考书
1.
2.
3.
4.
5.
6.
D S Kliger, J Lewis, and C Randall, “ Polarized Light in
Optics and Spectroscopy,” Academic Press, 1990, Chapters
4&5;
S Huard, “Polarization of Light,” John Wiley & Sons, 1996.
A Yariv and P Yeh, “Optical Waves in Crystals,” John Wiley
& Sons, 1984, Chapter 5;
H Kogelnik, R Jopson, and L Nelson, “Polarization-Mode
Dispersion,” Optical Fiber Telecomm IVB, 2002, Chapt 15.
Gordon and Kogelnik, “PMD fundamentals,” www.pnas.org.
C Poole and J Nagel, “Polarization effects in lightwave
systems,” Optical Fiber Telecom IIIA, I Kaminow and T
Koch, Eds. Academic, 1997, pp.114-161.
3
PMD: Polarization Mode
Dispersion
 Ideal single mode fiber: the two HE11 modes are degenerate. The
two orthogonal polarized modes have the same group delay.
 Real fibers contain some amount anisotropy owing to loss of
circular symmetry.
4
Origins of PMD I
5
Origins of PMD II

Optical Birefringence:

Intrinsic Perturbations:


Imperfections in the
manufacturing process:
noncircular core.
Extrinsic Perturbations:


Lateral stress, bending, or
twisting.
External environment
changes.
6
Evolution of Polarization in
Birefringent Fiber
• 理想的单色光；
• 脉冲。
7
Time-Domain PMD Effect in
Short Fibers

Pulse splitting due to
birefringence
 

ns
c

n f
c

n
c
Differential Group Delay
(DGD): group-delay
between the slow and
fast modes.

d  n  n  dn




L d  c  c
c d
DGD linear dependence of length
8
Beat Length Lb

Beat Length:



Lb 

n
SSMF: Lb~10m, n~10-7;
PMF: Lb~3mm;
Linear length dependence of DGD.
n  1
 
 DGD for a single beat length:  b  Lb
c
c 

b=5.2fs@1550nm.
9
Frequency-Domain PMD Effect


Output polarization
varies with frequency,
traces out a circle on
the Poincare Sphere.
Characteristic
Frequency
2
 cycle 

10
Polarization-Mode Coupling



Terrestrial and submarine transmission systems
~1000’s of km, Random variations in the axes of
the birefringence along the fiber length;
Concatenation of birefringent sections,
Polarization-mode coupling;
DGD increases with the square root of distance.
11
Correlation Length Lc





Any polarization state can be observed for a fixed input
polarization at large lengths due to random polarization
coupling;
Assuming <px>=1 and <py>=0 at the input, <px>-<py>
evolves from 1 to 0 at large lengths;
Lc as that length where the power difference has decayed to
<px>-<py>=1/e2;
Fiber transmission distance L<<Lc, the fiber is the shortlength regime, DGD increases linearly with distance.
Fiber transmission distance L>>Lc, the fiber is the longlength regime, DGD increases with the square root of
distance.
12
Frequency-Domain PMD
13
PSP: Principal States of
Polarization

Concatenation of birefringent
sections with birefringence axes
and magnitudes that change
randomly along the fiber.

PSP: two special orthogonal
polarization states at the fiber
input that result in an output
pulse that is undistorted to first
order.
14
Transmission Distances
Limited by PMD

PMD is an ultimate limitation for ultra-high speed
transmission systems!
15
光的偏振特性

光的偏振态（SOP）；

Jones矢量、Stokes矢量、Poincare球；

Jones矩阵、Mueller矩阵、Pauli矩阵；
16
State of Polarization (SOP) I

线偏振光
 

E  Ex  E y


 Ax i  Ay j sin(t  kz  o）
17
State of Polarization (SOP) II

圆偏振光

Ercp  Ao [sin(t  kz  o）i
 cos(t  kz  o）j ]

Elcp  Ao [sin(t  kz  o）i
 cos(t  kz  o）j ]
18
State of Polarization (SOP) III

椭圆偏振光

E  Ax cos(t  kz)i
 Ay cos(t  kz  ) j



方位角a(azimuth);
椭偏度e(ellipticity);
手性（handedness).
19
琼斯矢量与琼斯矩阵I



电场矢量

i
E  Ax eix iˆ  Ay e y ˆj
琼斯矢量
 Ax e i x 
V 
i y 
A
e
 y 
线性偏振
cos 
V 

sin






垂直线偏振态

X-线偏振光、Y-线偏振光
2
I 0  V  V  Ax2  Ay2
 sin  
V  

cos



20
琼斯矢量与琼斯矩阵II

圆偏振
1 1
ˆ
L

2 i 
 Rˆ 
1 1  i   Xˆ 
 ˆ 
1 i   ˆ 
2
L

Y 
 

变换矩阵
1 1
ˆ
R
 
2  i 
 Xˆ 
1 1 1   Rˆ 
 ˆ
i  i   ˆ 
2
Y

 L
 
Vcir  AVlin
1 1  i 
A


2 1 i 
Vlin  A1Vcir
1
A
2
1 1 
i  i 


21
琼斯矢量与琼斯矩阵III

椭圆偏振
 i 2


cos

e
ˆ
J (  , )  
i 2 
sin

e


 i 2



sin

e
ˆ
ˆ
J (  , )  J (    2 , )  
i 2 
cos

e


cosa cose  i sin a sin e 
ˆ
J (a ,  )  

sin
a
cos
e

i
cos
a
sin
e



变换矩阵
22
Stokes Parameters II

偏振光
tan   Ay Ax
tane  b a
2 Ax Ay
tan2a  2
cos
2
Ax  Ay
 s1  cos2e cos2a   cos2  
 s    cos2e sin 2a   sin 2  cos 
 2 
 

 s3   sin 2e
  sin 2  sin  
23
Jones Matrix

Linear interaction:

ˆ
ˆ

J  Ex i  E y j


 Ex  m11Ex  m12 E y
 

E y  m21Ex  m22 E y
 m11
M 
m21
m12 

m22 

J  Exiˆ  E y ˆj
 E   m11 m12   Ex 
 x   



 E y  m21 m22   E y 
k
M   Mi
i
24
琼斯矢量与琼斯矩阵I



电场矢量

i
E  Ax eix iˆ  Ay e y ˆj
琼斯矢量
 Ax e i x 
V 
i y 
A
e
 y 
线性偏振
cos 
V 

sin






垂直线偏振态

X-线偏振光、Y-线偏振光
2
I 0  V  V  Ax2  Ay2
 sin  
V  

cos



25
Stokes Parameters I

定义:
I0

 P0  
P   I  I 
y

 1   x
 P2   I  45o  I 45o 
   I I 
 P3   rcp lcp 
2
2
 P0   Ax  Ay 
 P   A2  A2 
y

 1   x
 P2  2 Ax Ay cos 

  
 P3   2 Ax Ay sin  
Si  Pi P0
 Ax e  i 2 
V 
i 2 
 Ay e 
(i  0,1...4)
26
Jones Matrix

Linear interaction:

ˆ
ˆ

J  Ex i  E y j


 Ex  m11Ex  m12 E y
 

E y  m21Ex  m22 E y
 m11
M 
m21
m12 

m22 

J  Exiˆ  E y ˆj
 E   m11 m12   Ex 
 x   



 E y  m21 m22   E y 
k
M   Mi
i
27
Jones Calculus of Birefringent
Crystal I
• o-xyz system:
Ax 

Jˆ   
 Ay 
 As   cos
A   
 f   sin 
sin    Ax 
A 

cos   y 
28
Jones Calculus of Birefringent
Crystal II
l 
 

0
 A   exp  ins c 
  As 


s
   
 
l   Af 

 Af  
0
exp  in f


c 

  ( ns  n f )
  ( ns  n f )
 A


0
  As 
i exp  i  2
s
   e 
 A 


0
exp
i

2
 Af 

 f 
29
l
c
l
2c
Jones Calculus of Birefringent
Crystal III
 A   cos
 x   
 Ay   sin 
 sin    As 
 

cos   Af 
 
 A 
 Ax 
x
    R(  ) M 0 R( ) 
 Ay 
 Ay 
M  R( ) M0 R( )
M M  1
30
Mueller Matrix

4x4 Matrix
S   MS
 m11
m
21

M
m31

m41
m12
m13
m22
m32
m23
m33
m42
m43
m14 
m24 

m34 

m44 
k
M   Mi
i
31
Partially Polarized Light

完全非偏振光
P1  P2  P3  0

部分偏振光
0 P P P  P
2
1

偏振光
2
2
2
3
2
0
2
2

Ax  Ay
 P0 
 P   A2  A2
x
y
 1  
 P2   2 Ax Ay cos
  
 P3   2 Ax Ay sin 
P P P P
2
1
2
2
2
3
2
0
32






Degree of Polarization (DOP)

偏振度(DOP)
DOP 
Ppolarized
Ppolarizaed  Punpolarize d
Ppolarized  P  P  P
2
1

2
2
2
3
偏振光 DOP=1; 部分偏振光 DOP<1; 完
全非偏振光 DOP=0.
33
Jones Vector and Stokes
Vector
s  sx , s y 

Jones Vector

Stokes Vector sˆ  s1, s2 , s3 
T
T
 s1  s x s x  s y s y



s

s
s

s
s
 2
x y
y x
s  j ( s s   s s  )
x y
y x
 3
34
Stokes Vectors of the Fundamental
States of Polarization
Xˆ
Yˆ
Xˆ  Yˆ
1 
1 
 
0
 
0
1
  1
 
0
 
0
1 
0
 
1 
 
0
cosXˆ
LCP
RCP
Elliptical
1 
0
 
0
 
1 
1


cos2e cos2a 


 cos2e sin 2a 


sin
2
e


 sin Yˆ
 1 
cos 2 


 sin 2 


0


1
0
 
0
 
  1
35
Poincare Sphere I
 s1  cos2e cos2a 
 s    cos2e sin 2a 
 2 

 s3   sin 2e

s  s  s 1
2
1

2
2
2
3
以s1, s2, s3为坐标的球;
36
Poincare Sphere II







赤道上的点表示线偏振;
北极点表示右旋圆偏振光;
南极点表示左旋圆偏振光;
同纬度点表示同椭偏度;
同经度点表示同方位角;
上半球表示右旋椭偏光;
下半球表示左旋椭偏光.
37
Polarization Evolution I
38
Polarization Evolution II
39
40
Jones Calculus of Birefringent
Crystal I
• o-xyz system:
Ax 

Jˆ   
 Ay 
 As   cos
A   
 f   sin 
sin    Ax 
A 

cos   y 
41
Jones Calculus of Birefringent
Crystal II
l 
 

0
 A   exp  ins c 
  As 


s
   
 
l   Af 

 Af  
0
exp  in f


c 

  ( ns  n f )
  ( ns  n f )
 A


0
  As 
i exp  i  2
s
   e 
 A 


0
exp
i

2
 Af 

 f 
42
l
c
l
2c
Jones Calculus of Birefringent
Crystal III
 A   cos
 x   
 Ay   sin 
 sin    As 
 

cos   Af 
 
 A 
 Ax 
x
    R(  ) M 0 R( ) 
 Ay 
 Ay 
M  R( ) M0 R( )
M M  1
43
Jones and Stokes Vectors I

Jones space
t T s
T e
 j0
U
T: transmission Matrix,
U: Jones Matrix.
 Stokes space:
tˆ  Rsˆ
ˆ
s,
ˆ
t,
s
t
T, U, R
IN
OUT
R: 3x3 rotation matrix in Stokes space.
44
Pauli Spin Matrix

Pauli spin matrix
1 0 
 0 1
0  j
 1  
  2  
  3  

 0  1
 1 0
j 0 

Hermitian and unitary
 i   i
Tr i  0
 i2  I
 i   i1
 i j   j i  j k
45
Birefringence Vector
Pauli spin vector in Stokes space

  ( 1,  2 ,  3 )
 Birefringence vector in Stokes space


  ( 1, 2 , 3 )

Birefringence vector in Jones space
 
    11  2 2  3 3
46
Jones and Stokes Vectors II
si  s  i s
 1 0   sx 
*
*
s1  s  1 s  (s , s ) 

s
s

s
s
 s  x x y y
 0 1  y 
*
x
*
y
s  s s
s s  s
47
Expansion of Matrix

Any 2x2 matrix M may be expanded
 m11 m12   a0  a1
  
M  
 m21 m22   a2  ja3
 a0 I  a1 1  a2 2  a3 3
 
 a0 I  a  
1
a0  Tr ( M );
2
a2  ja3 

a0  a1 
1
ai  Tr ( i M )
2
48
Connection between Jones
Vector and Stokes Vector

Unitary transmission matrix
TT   I ;

UU   I ;
RR  I ;
Write the components si of the Stokes
vector corresponding to s
si  s  i s

sˆ  s  s
49
Principal States of Polarization
(PSP)



Input polarization for which
the output state of
polarization is independent
of frequency to first order.
In the absence of PDL, the
PSPs are orthogonal.
The input and output PSPs
pˆ t  Rpˆ s
C Poole and R Wagner, “Phenomenological approach to polarization dispersion in
long single-mode fibers”, Elect Lett 22 1029 (1986),
50
PMD Vector

PMD vector in Stokes
space:

  pˆ

 : DGD; pˆ : the
direction of the slower
PSP.


 t  R s
51
Second-Order PMD Vector
PMD vector varies with
frequency;
 For large signal bandwidths



 (0   )   (0 )    (0 )    


Second-order PMD

d

 
   pˆ  p
d

52
Jones Matrix Eigenvector
Analysis I
Input:
Ea  e ae ja ( ) s
Output: Eb  e be jb ( ) t
IN


Eb ()  T ()Ea ()
T  e  j ( )U
dt
d
Q  jUU
t
s
T, U
OUT
  jQ  I  t  (1)

 d
e b db 
(2)
  
j 
e b d 
 d
53
Jones Matrix Eigenvector
Analysis II
dt
d
 0;
• Unitary matrix:
• Eigenvalues:
• Eigenvectors:
 j Q  I  t  0(3)
 u1 u2 
;
U   

  u2 u1 
  
u1
2
 u2
u1  u2  1
2
2
2
 u2  iu2 

u1  iu1 
p  e 
, 

D
D




i
54
Differential Group Delay




p p  0
Orthogonal:
e b e b  Im   0
No PDL:
Group delay:
2
2
b     Re   u1  u2
Differential group delay:
        2 u1  u2  2 det U
2

PMD vector:
2

  pˆ (4)
55
Pauli Spin Matrix Expansion



Q  Q
Hermitian matrix: Q  jUU ;
The trace is zero and the eigenvalues are real.
Expansion:
 2  j 3  1  
1  1

     (5)
Q  jU U  
1  2
2  2  j 3
jU U


1
p  p;
2
 
    p   p
  (1 ,  2 ,  3 )
56
Components of PMD vector

U is unitary

DGD

 u1
U   
  u2
u2 


u1 
  2 u1  u2  2 det U
Components  i
2
2
  1  2 j (u1u1  u2 u2 )

 2  2 Im(u1u2  u2 u1 )
  2 Re(u u  u  u )
1 2
2 1
 3
57
Jones Matrix for
PMD

Jones Matrix
e  j
D
 0
2
st
1
and
nd
2
U  Q1DQ
0 
j 2 
e 
s
t
Q
IN
D
Q
OUT
 cosk sin k 
Q

 sin k cosk 
H Kogelnik et al., “Jones matrix for second-order polarization mode dispersion”,
Opt Lett 25(1), 19 (2000),
58
Outline
Introduction
 Origins of Polarization Mode Dispersion
(PMD)
 Techniques of PMD compensation
 Programmable PMD compensator
 Results

59
Introduction

Transmission Loss


Chromatic Dispersion


Erbium-Doped Fiber Amplifier (EDFA)
Dispersion Compensated Fiber, Fiber
Grating, Optical Interferometers, etc.
Polarization Mode Dispersion (PMD)
Vary randomly
 ???

60
PMD Compensation: PSP
Transmission

The polarization of the
launched signal is along
one of PSPs at the fiber
input.
sˆ  pˆ s


Polarization control at the
fiber input, special
hardware at both the
transmitter and receiver.
Speed is limited.
61
PMD Compensation: PMD
Nulling


 fiber   comp

Adjustable birefringent
element.
Polarization controller.

Adjustable birefringence:




Opto-mechanical delay line;
Nonlinearly chirped PMfiber Bragg Gratings;
Variable delay line.
62
PMD Compensation: Fixed
DGD


Adjustable polarization controller
A single element with fixed DGD: a PM fiber of a fixed length.
63
PMD Compensation:
Programmable DGD

Birefringent crystal: YVO4








Large birefringence n=0.2 at 1.55 microns.
High optical quality, low loss.
High speed Polarization Controller
Low insertion loss: IL<3dB.
86 ps dynamic range
6 bit Resolution of 1.34ps
High speed~150 ms.
Programmable.
64