Transcript Nonlinear localization of light in disordered optical fiber arrays Alejandro Aceves* University of New Mexico AIMS Conference, Poitiers, June 2006 Research supported by the US.

Nonlinear localization of light in
disordered optical fiber arrays
Alejandro Aceves*
University of New Mexico
AIMS Conference, Poitiers, June 2006
Research supported by the US National Science Foundation
*Work in collaboration with Gowri Srinivasan
Outline



Motivation
The Optics Model
Proposed Work
Motivation

Study the phenomenon of light localization in optical
fiber arrays as a result of deterministic and random
linear and nonlinear effects, although as an
approximation the relevant randomness is only on the
linear part of the model.
The Optics Problem
2-D Hexagonal Optical Fiber Array
Governing Equation
Consider the nonlinear wave equation
n2
 tt    NL  0
2
c
(*)
The solution is given by
  A( x, y, z)ei ( kzt )
 tt  (i) 2 A( x, y, z)ei(kzt )
  2x, y A  Azz  2(ik ) Az  (ik ) 2 A ei(kzt )


Ignoring the non-linearity and substituting into the wave
equation (*), we have
 n2 2
 i ( kzt )
A
2
2


A

k
A

2
ik


A
0
x, y e
 2
z
 c

Governing Equation (contd.)
We substitute the following representations into equation (**)
k2 
2
c2
n2  1   f1 ( x  xm'n' , y  ym'n' )   n2 A f 2 ( x  xm'n' , y  ym'n' )
2
m 'n '
m 'n '
which gives

2 

2
A
f
(
x

x
,
y

y
)


n
A
f
(
x

x
,
y

y
)
1

m 'n '
m 'n '
2
2
m 'n '
m 'n ' 
c 2  m'n'
m 'n '

A
 2ik
  2x , y A  0
z
Governing Equation (contd.)
Substituting the following expression for the envelope A
A   amn ( z)U ( x  xmn , y  ymn )
m, n
Also at each core m,n we have the transverse mode
2
c2
f1 ( x  xmn , y  ymn )U mn   2x , yU mn   mnU mn
Substituting back into the wave equation and multiplying by
UNM, we integrate over x,y around the M,N fiber only considering
the terms in the summation for m’,n’ =M,N and m,n = M,N and
neighbors since U, f1 and f2 are local.
Governing Equation (contd.)
The final equation for the propagation of light in the fiber m,n
can be described as
2
 d
i dz   m,n   am,n  am,n 
cmn am1,n  am1,n  am,n1  am,n1  am1,n1  am1,n1   0
Writing the complex amplitude as ak  xk  iyk
real ODEs for each fiber


dxk
  y k    ( xk2  y k2 )  c y j
dz
j

we have two

dyk
 xk    ( xk2  y k2 )  c x j
dz
j
Numerical Solution of the ODE

We solve a system of 14 ODEs using Matlab

Light is initially input through the fiber in the middle.

We observe that due to symmetry, the 6 surrounding
fibers behave identically.

Assuming no losses, we have that the total energy is
conserved.
Numerical Solution (contd.)
Stochastic Model

Due to manufacturing imperfections the coupling and
propagation constants vary stochastically about a mean, with a
correlation function proportional to a delta function.
C  C0  C

   0  
Now the stochastic differential equations take the form of a
Langevin equation.


dxk
  y k    ( xk2  y k2 )  c y j  k y k  c j y j
dz
j
j


dyk
 xk    ( xk2  y k2 )  c x j   k xk   c j x j
dz
j
j
Langevin Equation

For N stochastic variables {X }  X 1 ,....,X N , the
general Langevin equations have the form
.
X i  hi ({X }, t )  gij ({X }, t )j (t )
where  j (t ) are Gaussian random variables with zero
mean and correlation function proportional to the
delta function and h and g are deterministic
functions.
Langevin Equation (contd.)
 The Drift and Diffusion coefficients Di and Di , j are calculated
as follows :

Di  hi ({X }, t )  g kj ({X }, t )
g ij ({X }, t )
X k
Di , j  g ij ({X }, t ) g ij ({X }, t )

W m S i

0
The Fokker-Planck equation may be written as
t
i 1 C i
where W is the probability density and the probability current
is defined by
S i  DiW  
j
DijW
C j
Moment Analysis



Given the Fokker-Planck equation, we can write the
second moments for all combinations of xi and yj
The equations for the average intensities form a
closed second order system.
It can be seen that the sum of the average intensities
is a constant, which agrees with the assumption of a
conservative system.
Numerical Solution of the SDE
 The non-linear SDE can be split into a linear and a non-linear
part:
da
i k  ( L  NL )a k
dz
 The randomness only occurs in the linear part, the non-linear
portion is deterministic.
 Formally, the solution for step z is
ak ( z  z)  exp[iz( L  NL)]ak ( z)
which can be approximated as
 zL 
 zL 


exp i
exp

i

zNL
exp
i
ak ( z )



2 
2 


Numerical Solution of the SDE
 The error involved in this approximation comes from the linear
part and since the operators do not commute, this error is of
the order of (z ) 3
 The linear steps are solved using an implicit midpoint method
which conserves quadratic invariants, in this case, the total
amplitude.
 The nature of the non-linear portion of the equation allows for
an exact solution of each equation in its complex form.
 Hence the total error remains the same as that in integrating
the linear part and the total amplitude (energy) is conserved
since each step conserves energy.
Numerical Solution of the SDE
(contd.)
Results

In order to study the localization phenomenon, the Monte Carlo
method is used to randomly sample the amplitude ratio of the
middle fiber to the total amplitude at different propagation
lengths within a period’s length.

A histogram is constructed for the amplitude ratios from 0 to 1
in intervals of 0.1.

Localization is said to occur if the amplitude ratio is skewed
towards 1.

We see that localization is observed to occur at higher
amplitudes.
Numerical Solution of the SDE
(contd.)
Numerical Solution of the SDE
(contd.)
Proposed Work – Continuum
Approximation of the Fiber Array

We intend to approximate the model of the discrete fiber array
as a continuum, with a broad distribution of light as shown in
the 1D figure below.
an
d
LB

LB
 1
d
We can make the following approximations
an ( z)  A( z, x)
A d 2  2 A
an1 ( z )  A( z, x  d )  A( z, x)  d

x 2 x 2
Proposed Work – Continuum
Approximation (contd.)

Substituting for the coupling terms, we have
2 A
an1 ( z )  an1 ( z )  2 A  d
x 2
2

This reduces the governing equation to a 1D Nonlinear
Schrodinger Equation
2
~ x
dA
2
 
2  A
i
 A   A A  2C0  C0 d


 A
2
dz
x
 

We will extend the continuum approximation to 2D
References




The Fokker Planck equation, H.Risken
Numerical solution of stochastic differential
Equations, Kloeden & Platen
Handbook of stochastic methods, Gardiner
Nonlinearity and disorder in fiber arrays,
Pertsch et al, 2004