Spiral Attractor Created by Vector Solitons
Sergey V. Sergeyev1, Chengbo Mou1, Elena Turitsyna1, Alexey Rozhin1, Sergei K. Turitsyn1, and Keith Blow1
1
Aston Institute of Photonic Technologies, Aston University, Birmingham, B4 7ET, UK
SUPPLEMENTARY INFORMATION
In the paper “Spiral Attractor Created by Vector Solitons”, we have demonstrated theoretically a new type of
chaotic polarisation attractor, which is found to be beyond the standard models of mode-locked lasers based on
coupled nonlinear Schrödinger or Ginzburg-Landau equations. Theoretical analysis has been done on the basis
of a new model, the derivation of which is presented in this Supplementary paper.
Vector model of erbium doped fibre laser mode locked by carbon nanotubes. The scheme of energy levels
for erbium doped silica is shown in Figure S1.
4
4
I13/2
3
4
t310 ms
I11/2
2
1
t210 ms
5
4
I15/2
Supplementary Figure S1: Three-level model of erbium doped fibre laser. (1) Pump at 980 nm, (2) non-radiative phononassisted transitions to the first excited level, (3) fluorescence from the first excited level at 1550 nm; (4) and (5) stimulated
emission and absorption at 1550 nm.
Semi-classical equations for a unidirectional laser can be written as followsS43, S44:

corresponding author email: sergey.sergeyev@gmail.com
E x
E
 c x  kEx  ik  e x Pg dg ,
t
z
E y
E y
c
 kEy  ik  e y Pg  dg ,
t
z
Pg 
   p  i 0 Pg   i p Dg m *e E x e x m e   E y e y m e ,
t
Dg 
i


*
  d  D0  Dg   Pg  E x e x  E y e y  Pg  E x*e x  E *y e y .
t
4














(1)
Here k, γp, γd are relaxation rates of photons in the cavity, P(g) and D(g) are medium polarization and normalized
gain angular distributions, g=(,,) are Euler angles describing the orientation of the local reference frame
(X',Y',Z') related to the Er3+ ion with respect to the laboratory reference frame (X,Y,Z) related to the orientation
of
the
cross-polarized
components
 ...dg  1 8  0 0 0 ...sin ddd
2 2 2 
of
S45
the
electric
field
ex
and
ey
in
the
EDFL.
Hence,
, E=Exex+Eyey is a lasing electric field, me is a unit vector along the
dipole moment of the transition with emissionS43, S44 , D0 is the scaled parameter of the pumping, Δ0 is detuning
of the lasing wavelength with respect to the maximum of the gain spectrum, the vector me* denotes complex
conjugation of me.
The relaxation rate of the medium polarization in erbium doped silica matrix γp= 4.75 x 1014 s-1 >> γd, k (γd=0.11 s-1, k=107 - 108 s-1)
S46
. This allows us to consider the limit P( g ) t  0 and so P(g) can be removed from
Eqs. (1) as follows:
1  i  D E  D E ,
Ex
E
 c x  kEx 
xx x
xy y
t
z
1  2
E y
E
1  i  D E  D E ,
 c y  kEy 
yx x
yy y
t
z
1  2
Dg 
D g 


  d  D0  Dg  
R Ex , E y , g ,
t
2


2
2
1  2
2
R Ex , E y , g 
Ex exm e  E y e ym e  Ex E *y exm e  e ym*y  E y Ex* e ym e e xm*e .

1  2 












(2)

Here    0  p and



Dxx  k  Dg  exme dg, Dxy  k  Dg  e yme e xm*e dg,
2



Dxy  k  Dg  e yme e xm*e dg, Dyy  k  Dg  e yme dg.
2
(3)
For erbium ions, we account for absorption from the ground state and pump wave polarization and saturation
according to Ref. 31 and so the equations for Dij (i,j=x,y) take the following form:
Dxx 
D yx 
1
2
  ng e m
x
1 
2
2
e

dg  1 , Dxy 


1 
2
*
 ng e x m e  e y m e dg , D yy 
 ng e y m e e x m e dg ,
*
(4)
2
1 
 ng  e y m e dg  1.
 
2 

Here n(g) is the angular distribution of the erbium ions at the first excited level which is determined by:
I
2
ng 
  d  p 1  ng  e pm a  ng   ng   1R Ex , E y , g

t
 I ps


,

2
E 2
*
Ey
Ex E y
E y Ex*
2
1  x
2
*


R Ex , E y , g 
e
m

e
m

e
m
e
m

e ym e e xm*e
x e
y e
x e
y e
1  2  I ss
I ss
I ss
I ss







Here 1   a L  is the EDF absorption at the lasing wavelength, I ps   d A  a( p ) p ,




(5)

.




I ss   d A  a( L) L are

saturation powers at the pump and lasing wavelengths    a( L)   e( L)  a( L) ,  a( L( e)) ,  a( p ) are absorption and
emission cross sections at the lasing wavelength and absorption cross section at the pump wavelength, ΓL and Γp
are the confinement factors of the EDF fibre at the lasing and pump wavelengths, ρ is the concentration of
erbium ions, A is the fibre core cross section area.
By adding a saturable absorber (single-wall carbon
nanotubes) and accounting for fibre birefringence, Kerr nonlinearity and chromatic dispersion
1-11
, equations (2)
and (3) take the following form
2
E x
E
 2 Ex
2
1
2


 i E x   x  i 2
 i  E x E x  E y E x  E y2 E x*   Dxx E x  Dxy E y ,
z
z
3
3
T 2


E y
z
 iE y  
1 1  i 
Dxx 
2 1 
2
E y
z
 i 2
2Ey
T
  ng e m
x
1  1  i 

2
2
e
2
2
1
2


 i  E y E y  E x E y  E x2 E *y   D yx E x  D yy E y ,
3
3



dg  1   2  N g  e x μ a dg   4 ,




2





*
*
 ng  e y m e e x m e dg   2  N g  e y μ a e x μ a dg,
2 1  2
  1  i 
*
*
D yx  1
 ng e x m e  e y m e dg   2  N g e x μ a  e y μ a dg,
2 1  2
2
2
 1  i  
D yy  1
 ng  e y m e dg  1   2  N g  e y μ a dg   4 ,.
2  


2 1  
Dxy 
 Ip
2
ng 
1  ng  e p m a  ng   ng   1c3 REr E x , E y , g
d
 I ps
t


N g  
REr




2
2

Ey
E x E *y
E y E x*
2
1  Ex
2
*


Ex , E y , g 
e
m

e
m

e
m
e
m

e y m e e x m *e
x
e
y
e
x
e
y
e
I ss
I ss
I ss
1  2  I ss


RCNT
1   3 RCNT
1
E x , E y , g 1  2

,





2
2

Ey
E x E *y
E y E x*
2
1  Ex
2
*


Ex , E y , g 
e
μ

e
μ

e
μ
e
μ

e y μ a e x μ *a
x a
y a
x a
y a
I ss
I ss
I ss
1  2  I ss









.


,


(6)
Here N(g) is the angular distribution of the CNT in the ground state,  2 is the CNT absorption at the lasing
wavelength, α3 is the ratio of saturation powers for CNT and EDF, α4 represents the normalized losses, β is the
birefringence strength (2 β=2π/Lb , Lb is the beat length), μa is a unit vector along the dipole moment of the
transition with absorption for CNT, T=t-z/Vg, Vg is the group velocity, η=βλ/(2πc) is the inverse group velocity
difference between the polarisation modes.
We introduce a new slow-time variable t s  z Vg t R  , where tr=L/Vg is the photon round-trip time, L is the cavity
length) and assume an ansatz in the form:
E x (T , t s )  u(t s ) sech(T Tp ), E x (T , t s )  v(t s ) sech(T Tp ).
(7)
Here Tp is the pulse width. After substitution of (7) into (6) and averaging over the time Tp<<T<<tR we obtain
the following equations:
u
2 2
1
 2

 iLu  i 2 Lc1u  iI ss Lc 2  u u  v u  v 2u *   Dxxu  Dxyv,
ts
3
3


v
2 2
1
 2

 iLv  i 2 Lc1v  iI ss Lc 2  v v  u v  u 2v*   Dyxu  Dyyv,
ts
3
3


1L 1  i 
Dxx 
2 1 
2
  ng  e m
1L 1  i 
x

2
e

dg  1   2 L  1   3c3 RCNT  e xμ a dg   4 L,
2














*
2
*
 ng  e ym e e xm e dg   2 L  1   3c3 RCNT 1   e yμ a e xμ a dg ,
1  2
 L 1  i 
*
2
*
Dyx  1
 ng e xm e  e ym e dg   2 L  1   3c3 RCNT 1   e xμ a  e yμ a dg ,
2 1  2
2
2
 L 1  i  
Dyy  1
 ng  e ym e dg  1   2  1   3c3 RCNT 1  2 e yμ a dg   4 ,
2  


2 1  
2
ng 
  W p 1  ng  e pm a  ng   ng   1c3 REr ,


ts
Dxy 
2

REr 
RCNT



(8)




2
1  2
2
2
u e xm e  v e ym e  uv* e xm e  e ym*e  vu* e y m e e xm*e ,

1  2 
2
1  2
2
2
Ex , E y , g 
u e xμ a  v e yμ a  uv* e xμ a  e y μ*a  vu* e yμ a e xμ*a 

1  2 







Here ε=tRγd and u, v are normalized to the saturation power Iss and Wp is normalized to the saturation power Ips.





We have also used the approximation  3c3 R Ex , E y , g 1  2 e x μ a  e y μ*a  1 and have neglected the inverse
group velocity difference corresponding to   0 . For a cavity length Lc=7.8 m, beat length Lb=5m, and
λ=1.56 μm the time delay between cross polarised pulses over the length of the cavity can be found based on
notations to Eqs. (6) as Td=4 fs. The time delay is much less then the pulse duration of 600 fs and the CNT
relaxation time of 300 fs and so the group velocity difference can be ignored in Eqs. (6). The coefficients ci
(i=1,2,3) are defined as
coshx   2
dx

3
1 T T p coshx 
c1  2
 0, c2 
T Tp
Tp
 sechx dx
T Tp
T Tp
2
T Tp
 sechx  dx
3
T T p
 sechx dx
T T p
1
,c3 
2

T Tp
2
 sechx  dx
T T p
T T p
T Tp
 sechx dx

2

(9)
.
T T p
To simplify equations (8) further, we use an approximation the validity of which has been justified in Ref.
33
,
viz. we assume that dipole moments for erbium and CNT are located in the plane (ex, ey) as shown in Fig. 3 32, 33.
We
now
make
pump e p  e x  ie y 
the
approximation
ma=me
and
consider
an
elliptically
polarized
1   2 (here δ is the ellipticity of the pump wave)
m ee x   cos , m ee y   sin , m a e p 2  cos 
  2 sin 
,
1  2
2
μ a e x   cos1 , μ a e y   sin1 
2
(10)
The angular distribution n(g) now depends only on θ and so can be expanded into a Fourier series as follows
32,33
:
n  

n0 
  n1k cosk    n2 k sin k .
2 k 1
k 1
(11)
As a result, we find a complete set of equations for Ex, Ey, n0, n12, n22 as follows:
u
LI ss  2
2 2
1 2 *
 i Lu  i
 u u  v u  v u   Dxxu  Dxyv,
ts
2 
3
3

v
LI ss  2
2 2
1 2 *
 iLv  i
 v v  u v  u v   Dyxu  Dyyv,
ts
2 
3
3

  L1  i 

  L1  i 

Dxx   1
I xx (n0 , n12 , n22 )  J xx   4 L , Dxy  Dyx   1
I xy (n0 , n12 , n22 )  J xy ,
2
2
 1 

 1 

  L1  i 

Dyy   1
I yy (n0 , n12 , n22 )  J yy   4 L 
2
1




n
n
n
 n

 n

I xx (n0 , n12 , n22 )    0  1   12 , I yy (n0 , n12 , n22 )    0  1   12 , I xy (n0 , n12 , n22 )   22 ,
2
2
2
2
2




2
2
2  L
2
2 
2
2 
1
1
J xx   2 L   3
3 u  v , J yy   2 L   3
u  3 v , J xy   3 2 uv*  vu* ,
2
8

2
8

8











I p 1 2

 Ip

dn0
   I p  2 R10  1   R10 n0   R11 
dts
2
2 1  2












 n
 
12



 n22 R12 ,

 1 2 I p
 1 2 I p
n 
 Ip

dn12

 R11    1  R10 n12  
 R11  0 ,
2
2
dts
1


2
2
1


2




 2 

 Ip

dn22
n 
   R12    1  R10 n22  R12 0 ,
dts
2 

 2

1
1
1
2
2
2
2
R10 
u  v , R11 
u  v , R12 
uv*  vu* ,
 1  2
 1  2
 1  2




The equations for Ψ  u, v T can be presented in the form





(12)
Ψt s  1  exp( G)Ψt s ,
(13)
where
t s 1
  Dxx t s dt s
G  t st s1

  Dxy t s dt s
 ts

t s 1
 Dxy t s dts 
,

 D yy, t s dts 
ts

ts
t s 1
(14)
If an in-cavity polarisation controller is installed then equation (12) can be modified as follows:
Ψt s  1  T exp( G)Ψt s ,
(15)
Where T is the transfer matrix of the polarisation controller 37
 A  i B C  i D
T
,
 C  i D A  i B 
A   cos  1 cos  2 , B   sin  3 sin  1 ,
(16)
C   cos  1 sin  2 , D   sin  1 cos  3 ,
 1       / 2,  2   2 ,  3   2   ,
Here α/2, γ/2, and (α+ξ)/2 are the orientations of the first quarter-wave plate (QWP), half-wave plate and the
second QWP with respect to the vertical axis Y.
By using a vector form of equations (9) we derive the following differential equations:

dΨ  dG

 ln(T) Ψ,
dt s  dt s

(17)
Phase and amplitude anisotropy caused by fibre birefringence and polarization controller can be accounted for
as follows:
iL 0   a  i b
ln(T)  

 0  i L    c  i d
c  id 
.
a  i b 
(18)
Thus, equations (12) for Ex, Ey can be rewritten as
u
LI ss  2
2 2
1 2 *
i
 u u  v u  v u   Dxx  a  i b u  Dxy  c  i d v,
ts
2 
3
3



v
LI ss  2
2 2
1 2 *
i
 v v  u v  u v   Dyx  c  i d u  Dyy  a  i b v,
ts
2 
3
3


 

(19)
Supplementary References
S43 H. Fu, H. Haken, Semiclassical dye-laser equations and the unidirectional single-frequency operation, Phys.
Rev. A 36, 4802-4816 (1987)
S44 S.V.Sergeyev, Orientational-relaxation dependent bichromatic operations of a ring cavity dye laser with
polarized pumping, Opt. Comm. 131, 399-407 (1996).
S45 D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum
(Singapore: World Scientific, pp. 21-23, (1988).
S46 Q. L. Williams and R. Roy, Fast polarization dynamics of an erbium-doped fiber ring laser, Opt. Lett. 21,
1478-1480 (1996).
Figure captions
Supplementary Figure S1: Three-level model of erbium doped fibre laser. (1) Pump at 980 nm, (2) non-radiative phononassisted transitions to the first excited level, (3) fluorescence from the first excited level at 1550 nm; (4) and (5) stimulated
emission and absorption at 1550 nm.