Supplementary Information
Stable nonlinear optical vortices in multicore fiber
A. Model equations
General equations
M
A
i 0   0 A0   C0 m Am  2 0 | A0 |2 A0  0,
z
m1
(1)
Am
2
i
  m Am  Cm 0 A0  Cm,m1 Am1  Cm,m1 Am1  2 m | Am | Am  0, m  1,..., M
z
We will start the linear stability analysis from the following equations:
-all periphery cores are identical C m,m 1  C m,m 1  C1 ,  m   1 ,  m  1 , m  1,.., M
-central core can be different from periphery cores: C0 m  C0 , m  1,..., M
M
A0
  0 A0  C 0  Am  2 0 | A0 | 2 A0  0,
z
m 1
A
i m  1 Am  C 0 A0  C1 ( Am 1  Am 1 )  2 1 | Am | 2 Am  0, m  1,..., M
z
i
(2)
This system supports vortex solutions with topological charge S, which is characterized
by amplitude pattern in cores:
A0  0, Am  Ae
i
2S
m
M
e iz , m  1,..., M
(3)
where the amplitude A can be calculated from the Eq. (2):
A2  
  1  2C1 cos 
2
(4)
The vortex solutions exist for
  (1  2C1 cos )
Depending on S and number of periphery cores M, parameter ranges with stable vortices
are found analytically and/or numerically.
B. Liner stability analysis: General case MCF + central core
i
2S
m
iz
M
 iz
e
Introducing small perturbations Am  ( A  Am )e
and A0  (0  A0 )e on
the top of the vortex solutions (8) in the Eq. (2), after the linearization the following
eigenvalue problem is obtained:
M
A0
 (    0 )A0  C0  Am e im  0,
z
m1
Am
i
 (   1 )Am  C0A0 e im  C1 (Am1e i  Am1e i )  2 1 A2 (2Am  Am* )  0,
z
m  1,..., M ,
2S

M
(5)
i
The equations (5) and corresponding complex conjugate can be written in a matrix form:

A
 iM 1A ,
z
(6)
where A is column matrix A | A0 , A1 ...AM , A0* , A1* ...AM* |T , and the corresponding

eigen matrix M 1 has a form


F
M1   *
G

G

 F*
(7)


where F and G are block matrixes each with dimensions ( M  1)  ( M  1) :

F
0
C 0 e i 
C 0 e  2 i
C 0 e i
1
f*
C0 e i 2
f
1
.
.
0
f
.
0
0
C0 e i ( M 1)
C 0 e i  M
... C0 e i ( M 1)
...
0
...
0
...
...
...
1
f*
C 0 e i M
f*
0
f
1
(8)
( M 1)( M 1)
0     0
1    1  4 1 A2
f  C1ei
(9)
0 0
0 ... 0
0
0 g
0 ... 0
0
 0 0
G
. .
g ... 0
. ...
0
0 0
0 ... g
0
0 0
0 ... 0
g ( M 1)( M 1)
g  2 1 A2
(10)
(11)
In a general case the matrix is not Hermitian one, and the type of the eigenvalues (real stability, pure imaginary – exponential instability or complex – oscillatory instability)
depends on the values of the matrix elements, and consequently on the parameters
(  0 ,  0 , 1 ,  1 , A, S , M ). The eigenvalue problem is solved numerically and
corresponding eigenvalues are calculated and stability properties are discussed.
C. Linear stability analysis: MCF without central core
The corresponding matrix equation reads:

A
 iM 2A
z
(12)

where A is column matrix A | A0 , A1...AM , A0* , A1* ...AM* |T and M 2 is the
corresponding eigen matrix.

The matrix M 2 in this case can be written in a form



F
G
M2  

 G  F*


where F and G are block matrixes now each with dimensions ( M  M ) :
(13)
1
f*

0
F
.
0
f
...
0
f*
1 f ...
f * 1 ...
.
. ...
0
0
0
0
... 1
... f *
f
1
f
0
0
0
0
0
(14)
( M M )
f  C1 exp( i )
2S

M
1    1  4 1 A2


G  2 1 A 2 I - is the diagonal matrix with dimension M
This form corresponds to the introduced perturbations in a form
 iz
im  iz
( A0 e , Am  ( A  Am )e e , m  1,..., M , )
The eigenvalues i , i  1...2M are the solution of the equation




F  I
G
=0
(15)
det( M 2  I 2 M )  det


G
 F *  I


It is easy to prove that G and F  I commute and then the following rule for the block
matrix holds

F  I
det

G

G




 
 det[( F  I )(  F *  I )  GG]   det{FF *  [ ( F  F * )  2  4 12 A 4 ]I }
*
 F  I
(16)


 
Because the matrix F is circulant then FF * and F  F * are also circulant matrices and


 
hence the whole matrix M c  FF *  [ ( F  F * )  2  4 2 A 4 ]I is also circulant.

Hence, the matrix M c is completely defined with the vector containing only the first row.

The elements ( h j , j  0,..., M  1 ) of the first row of the matrix M c can be calculated:
h0  12  2  4 12 A 4  2c12 cos( 2 )
h1  21c1 cos( )  2ic1 sin(  )
h2  c12
h3  h4  ...  hM 3  0
hM 2  c12
hM 1  21c1 cos( )  2ic1 sin(  )
(17)

Based on the properties of the circulant matrix the  k of the matrix M c are:
 j  12  2  4 12 A4  2c12 cos(2 )  41c1 cos( ) cos(j)  4c1 sin(  ) sin( j )  2c12 cos(2j )
(18)
2
where  
M

The all 2M eigenvalues of the marix M 2 can be calculated from the equations
j 0
which has a form
2  2a  b  0
(19)
where:
a  2c1 sin(  ) sin( j )
b  12  4 12 A4  2c12 cos(2 )  41c1 cos( ) cos(j )  2c12 cos(2j )
(20)
(21)
The stability condition is d  a 2  b  0 .
Using the definition 1    1  4 1 A2 and the vortex solution
  1  2C1 cos 
A2  
2
and some basic trigonometric identities, the discriminant d  a 2  b takes a simple form:
d  8 1 A2c1 cos( )(1  cos(j ))  4c12 cos 2 ( )[1  cos(j )]2
(22)
The stability condition d  0 leads to:
2
1. cos( )  0 and A 
c1
cos( )[1  cos(j )] . The condition cos( )  0 leads to
2 1
the condition

2
 2n   
4n  1 S 4n  3
2S 3



 2n , or
4
M
4
M
2
All vortices which satisfy the condition (23) are stable.
(23)
According to (23) follows that stable vortices are:
For S=1, stable M=3,4
For S=2, stable M=3,4,5,6,7,8
For S=3, stable M=4,5,6,7,8,9,10,11,12
For S=4, stable M=3,6,7…16
….
This analytical results are in full agreement with the numerical results.
2
2. cos( )  0 and A 
c1
cos( )[1  cos(j ] . The condition cos( )  0 leads to
2 1
the condition
4n  1 S 4n  1


.
4
M
4
So, stable vortices must satisfy condition (24) simultaneously with the condition
c
A2  1 cos( )[1  cos(j ] .
2 1
(24)
(25)
In another words, all vortices satisfying condition (24) are unstable except in the narrow
low power stability region defined by (25).
For S=1, M=5,6,7….
For S=2, M=9,10…
For S=3, M=3, 13, 14….
For S=4, M=4,5, 17,18,...
Again, the results and is a full agreement with the numerical modelling.
For linear case for MCF the discriminant (25) is always positive, and directly follows that
all vortices are stable independently on the amplitude.

Moreover for the linear case (MCF+central core) the matrix M 1 is reduced to


F
M 1L 
0
0
(26)

 F*

The matrix M 1L is Hermitian and all eigenvalues are real. Therefore, it is not necessary to
compute the eigenvalues to conclude that all vortices in linear MCF (with and without
central core) are stable.