Multi-scale polarisation phenomena
Vladimir Kalashnikov1, Sergey V. Sergeyev1*, Gunnar Jacobsen2, Sergei Popov3 and
Sergei K. Turitsyn1
1
AIPT, Aston University, B4 7ET, Birmingham, UK
2
Acreo, Electrum 236, SE-16440, Kista, Sweden
3
Royal Institute of Technology, SE-1640, Stockholm, Sweden
*
Contact: s.sergeyev@aston.ac.uk
SUPPLEMENTARY INFORMATION
In the paper “Multi-scale polarisation phenomena”, we have modelled a stochastic alteration
of statistical properties of polarisation phenomena in a fibre Raman amplifier and
demonstrated a transit from the polarisation pulling to the resonance-like escape from the
pulling. In this Supplementary paper we present an application of a widely used multi-scale
technique for derivation of coupled Manakov’s equations describing evolution of the state of
polarisations for pump and signal waves in a fibre Raman amplifier.
Multi-scaling technique
We start from the coupled Manakov-PMD equations describing co-propagation signal and
pump waves in a fibre Raman amplifier [1-5]:
i
 As
z
  s  cos    3  sin    1  As  i
s
As  i s'  cos    3  sin    1 
2
 As
t

 2 As

1
  s
  ss 2 As As  As* As* As 
2
2
t
3
2 sp
ig

2 Ap Ap  Ap Ap  A*p A*p As  R Ap Ap A s  0,
3
2
 Ap
 Ap
p
i
  p  cos    3  sin    1  Ap  i
Ap  i p'  cos    3  sin    1 

z
2
t
 2 Ap
 pp
1
(1)
  p

2 Ap Ap  A*p A*p Ap 
2
2
t
3
2 sp
ig  p

2 As As  As As  As* As* Ap  R
As As A p  0.
3
2 s








Where
0 1
0  i 
1 0 
,2  
, 3  
, Ai  Aix , Aiy T , Ai  Ai



1 0
i 0 
0  1
1  
By using rotation matrix R as follows [3-5]:


 Aix* , Aiy*

(2)
Ai  R ai ,
 cos / 2 sin  / 2
R
,
 sin  / 2 cos / 2
(3)
We find:
 2 as  ss
1
  s

2 as as  as* as*
z
2
t
2
t 2
3
2 sp
ig
2 a p a p  a p a p  a*p a*p as  R a p a p a s  0,
3
2
 ap
 ap
2 ap
p
1
'

i
  p ap  i
a p  i p 3
  p

z
2
t
2
t 2
 pp
2 sp

2 a p a p  a*p a*p a p 
2 as as  as as  as* as* a p 
3
3
ig  p
 R
as as a p  0.
2 s
i
 as
  s as  i
s
as  i s' 3
 as


A
s





(4)

Where

i   i
 z
z 

, z 
 g z , g z   0, g z g z '   2 z  z ' .

  i 
z



(5)
By using unitary transformation T [3-5]
U  T as ,
 t
T   1*
 t2
V  T ap ,
t2 
,
t *1 
(6)
t1  t2  1, t1 0  1, t2 0  0, i
2
2
T
  pT  0,
z
We find
U
2 U
1

i
   s   p   3 U  i
U  i  3
  s

z
2
t
2
t 2
2 sp

ig
 ss  2 U U U  N1s  
2 V V U  V V U  N 2 s   R V V U  0,

3
3
2
2
 ap
p
 pp
V
 V
1
i
i
V  i p'  3
  p 

 2 V V V  N1 p  
2
z
2
t
2
t
3
2 sp
ig  p

2 U U V  U U V  N2 p   R
U U V  0.

3
2 s
s
'
s
U
(7)
Here
 1
 4
~3  T  3T  
*4 
,
 1 


 b U 2U *  b U 2U *  b U 2U *  2U U 2  b U 2U *  2b U U 2 
5
x
x
2
y
x
3
y
y
y
x
1 x
y
4
x
y 
N1 s  
,
2
2
*
*
2
*
2
*
2
*
 b U U  b U U  b U U  2U U  b U U  2b U U 2 
5
y
y
2
x
y
1
x
x
x
y
3
y
x
4
y
x






 b U V
 b U V
 b U V 2  b U V V * U V 2 U V 2
5
x
x
3
x y x
y
x
x
y
N2s  
b U V 2  b U V V * U V 2 U V 2
1
y x y
x
y
x
y
 5 y y


4
x
4
y

 U yVxVy*  b2U yVyVx*  b1U xVxVy* 
,
2
 U xVyVx*  b2*U xVxVy*  b3U yVyVx* 
x
2

y
 b V 2V *  b V 2V *  b V 2V *  2V V 2  b V 2V *  2b V V 2 
5 x
x
2 y
x
3
y
y
y
x
1 x
y
4 x
y 
N1 p  
,
 b V 2V *  b*V 2V *  b V 2V *  2V V 2  b V 2V *  2b V V 2 
2 x
y
1
x
x
x
y
3 y
x
4 y
x 
 5 y y
N2 p




 
 

 b V U 2  b V U U *  V U 2  V U 2  b V U 2  V U U *  b V U U *  bV U U * 
5 x
x
3
x
y
x
y
x
x
y
4
x
y
y
x
y
2 y
y
x
1 x
x
y 

.
 b V U 2  b V U U *  V U 2  V U 2  b V U 2  V U U *  b*V U U *  b V U U * 
1
y
x
y
x
y
x
y
4
y
x
x
y
x
2 x
x
y
3 y
y
x 
 5 y y

(8)
Here
b1   3 6 , b2   *6  , b3   3*6 , b4   32 , b5   6 ,
2
2
1  t1  t 2 ,  2   t1t 2  t1*t 2* ,  3  i t1t 2  t1*t 2* ,
2
2
 4  2t t ,  5  t  t
*
1 2
2
1
 ,
* 2
2
6

 i t  t
2
1
 ,
(9)
* 2
2
1 0   1,  2 0   0,  3 0   0,  4 0   0,  5 0   1,  6 0   i,
12   22   32  1,  24   52   62  0,
And equations for Ωi take the form:
dΩ
 Qg ( z )  Λ,
dz
Q  Q1 , Q2 , Q3 , Q4 , Q5 , Q6 T , Λ  1 ,  2 ,  3 ,  4 ,  5 ,  6 T



g z   0, g z g z '   2 z  z ' ,
(10)
1  0,  2  2 p  3 ,  3  2 p  2 ,  4  0,  5  2 p  6 ,  6  2 p 5 ,
Q1  2 2 , Q2  21 , Q3  0, Q4  25 , Q5  2 4 , Q6  0.
For averaging over random birefringence we use the following notations [3-5]
6
d  i 

2 6 6 
Qi  
.
 G i  , G...    i
     QiQk  Qk
dz
i
2 i 1k 1
k i 
i 1
In further notations we skip brackets for averaging <…> and so obtain:
(11)
d1
d 2
d 3
 2 21 ,
 2 p  3  2 2 2 ,
 2 p  2 ,
dz
dz
dz
d 4
d5
d 6
 2 2 4 ,
 2 p  6  2 25 ,
 2 p 5 ,
dz
dz
dz
1 ,  2 , 3 ,  4 , 5 , 6 ,  0  1,0,0,0,0,0.

2
As a result: 1 z   exp  2 z


(12)
.

d12
 4 2  22  12 ,
dz
d 22
 4 p  23  4 2 12   22 ,
dz
d32
 4 p  23 ,
dz
d 2  3
 2 p  22  32  2 2 23 ,
dz
2
1 ,  22 , 32 ,  23 0  1,0,0,0.



(13)





d 24
 4 2 52   24 ,
dz
d52
 4 p 56  4 2  24  52 ,
dz
d62
 4 p 56 ,
dz
d56
 2 p 52  62  2 256 ,
dz
 24 , 52 , 62 , 56 0  0,1,1,i .






(14)
d 4
2
2
2
 4 2  5   4 ,


dz
d 5
2
dz
d 6


2
2
 2 p 5*6  6*5  4 2  5   4 ,


2


 2 p 5*6  6*5 ,
dz
2
2
d 5*6  6*5
 4 p  5  6   2 2 5*6  6*5 ,
dz






(15)


  2 ,  2 ,  2 ,  *   *   0, 1, 1, 0.
 4
5
6
5 6
6 5 0


d 36
 2 p  2 6   35 ,
dz
d  2 6  35 
 4 p  25   3 6   2 2  2 6   35 ,
dz
d 25
 2 p  2 6   35   4 2  25  1 4 ,
dz
d1 4
 4 2  25  1 4 ,
dz
36 , 26  35 , 25 , 14  0  0, 0, 0, 0.




d 3*6
 2 p  2*6   3*5 ,
dz
d 2*6
 2 p  2 *5   3*6  2 2 2*6 ,
dz
d 2*5
 2 p  2*6   3*5  4 2  2 *5  1*4 ,
dz
d1*4
 4 2  2 *5  1*4 ,
dz
d 3*5
 2 p  2*5   3*6  2 2 3*5
dz
 3*6 ,  2*6 ,  2*5 , 1*4 ,  3*5 0  0, 0, 0, 0,0.




(17)




(16)


As result we find:


1 z   exp  2 2 z ,b1  0, b3  0, b2  0, b4  0, b5  0.
To check a validity of averaging procedure, we consider the Raman amplification in a short
piece of fibre (5 km) with low input signal power (10 mW). These conditions allow us to
neglect pump depletion, cross-phase and self-phase modulation and to skip time dependence.
It has been found that margins of such approximation are: pump powers Pin <1 W, signal
powers Ps <10 mW and 𝐷𝑝 > 0.01 ps km-1/2 , pulse width > 2.5 ps for fibre length L <20 km
[1, 6].
With accounting for these notations and by using Jones to Stokes space transformation






 
S  U  U , P  V  V ,   i  1  j  2  k  3 , we arrive to

 0 



dS g R  
2


P S  PS    s S   s   p exp  2 z   S3 ,
dz
2
 S 
 2 


 
g
dP
  R p P S  PS    p P.
dz
2s

(18)

Here P  P  P12  P22  P32 , S  S  S12  S 22  S32 .
Numerical technique
The equations describing distributed Raman amplification with a gain coefficient g and a
distance-dependent pump and signal powers P0(z), s0(z), respectively, as well as the unit
vector components of pump and signal SOPs (p1,p2,p3) and (s1,s2,s3), respectively, can be
expressed in the form of system of stochastic differential equations of Stratonovich type
(Equation 7 in main text [1]):
P0  z   Pin exp   p L1    2  z ,
ds0 z  
1
gP0 z x z s0  z dz ,
2
1
d s0  z s1  z   gP0  z s0 z  p1  z dz  s 2 z s0 z   dW z ,
2
1

d s0  z s 2  z    gP0  z s0 z  p 2  z   2bs s3  z s0 z  dz  s1 z s0 z   dW z ,
2

1

d s0  z s3  z    gP0  z s0 z  p3  z   2bs s 2 z s0  z  dz ,
2

dp1 z   p 2 z   dW z , dp 2 z   2b p p3 z dz  p1 z   dW z ,
(19)
dp3 z   2b p p 2 z dz ,
1

d s0  z x z    gP0 z s0  z   2 s0 z b p  bs  p3 z s 2 z   p 2 z s3  z  dz 
2

 1   s0  z  p1 z s 2 z   p 2  z s1  z   dW  z .
Here η =+1 or -1for co- or backward pump-signal propagation, xz   pˆ  sˆ is a pump-tosignal SOP projection, and W(z) represents a Winer process with a drift 0 and volatility 1/Lc.
Different algorithms for solution of Equation 19 have been tested in framework of both
Wolfram Mathematica 9.0 and Matlab environments. Figure S1 demonstrates the sets of 10
stochastic trajectories with the corresponding average (dashed curve) values and standard
deviation ranges (filled) for a signal power evolution obtained on the basis of different
numerical techniques (Wolfram Mathematica 9.0).
Figure S1. Ten stochastic trajectories of the signal power s0. The orange dashed curve shows the averaged
values of s0. Filled domains demonstrate the ranges of the s0 standard deviation. The parameters correspond to
the point B and
p̂ =(1,0,0), ŝ =(1,0,0) in Figure 2 in the main text.
Our tests have demonstrated that only algorithms shown in bottom row in Figure S1 do not
produce destructively wandering stochastic trajectories which would lead to numerical
divergences as it takes a place for the algorithms shown in top row (Figure S1, the algorithms
shown in top row diverge). Additionally, the tests have demonstrated that the KloedenPlaten-Schurz algorithm provides a fastest convergence in our case.
The variation of step size z  10  n min L, Lc , Lbs  within a range of n=3÷5 suggested that
n=4 provides with a perfect numerical convergence for a whole diapason of PMDs
considered. One has to note that a relatively large step-size (e.g., n<3) shifts the <G> and G
curves into region of larger Dp. Also, we tested a sensitivity of results obtained to a number
of stochastic trajectories. We may conclude that 100 stochastic trajectories provides with a
sufficient statistical precision in our case (see Figure S2).
40
100 trajectories
50 trajectories
G
30
20
10
0
10-2
1/2
10-1
Dp, ps/km
Figure S2. RMS gain fluctuations
G
to the point B and
obtained for a different number of stochastic trajectories corresponding
p̂ =(1,0,0), ŝ =(1,0,0) in Figure 2 in the main text.
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