Nonlinear localization of light in disordered optical fiber

arrays

Alejandro Aceves*University of New Mexico

AIMS Conference, Poitiers, June 2006Research 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 EquationConsider the nonlinear wave equation

(*)

The solution is given by

Ignoring the non-linearity and substituting into the wave equation (*), we have

)(),,( tkziezyxA

02

2

NLc

ntt

)(2 ),,()( tkzitt ezyxAi )(22

, )()(2 tkzizzzyx eAikAikAA

02 )(2,

222

2

tkziyx eA

z

AikAkA

c

n

Governing Equation (contd.)

We substitute the following representations into equation (**)

which gives

2

22

ck

''

''''2

2

2''

''''12 ),(),(1

nmnmnm

nmnmnm yyxxfAnyyxxfn

''''''2

2

2''

''''12

2

),(),(nm

nmnmnm

nmnm yyxxfAnyyxxfAc

02 2,

Az

Aik yx

Governing Equation (contd.)

Substituting the following expression for the envelope A

Also at each core m,n we have the transverse mode

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.

),()(,

mnmnnm

mn yyxxUzaA

mnmnmnyxmnmnmn UUUyyxxfc

2

,12

2

),(

Governing Equation (contd.)

The final equation for the propagation of light in the fiber m,n can be described as

Writing the complex amplitude as we have two real ODEs for each fiber

j

jkkkk ycyxydz

dx)( 22

jjkkk

k xcyxxdz

dy)( 22

kkk iyxa

nmnmnm aadz

di ,

2

,,

01,11,11,1,,1,1 nmnmnmnmnmnmmn aaaaaac

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.

Now the stochastic differential equations take the form of a Langevin equation.

j

jjkkj

jkkkk ycyycyxydz

dx )( 22

CCC 0 0

j

jjkkj

jkkkk xcxxcyxxdz

dy )( 22

Langevin Equation

For N stochastic variables , the

general Langevin equations have the form

where are Gaussian random variables

with zero mean and correlation function

proportional to the delta function and h and g

are deterministic functions.

NXXX ,....,}{ 1

)()},({)},({.

ttXgtXhX jijii

)(tj

Langevin Equation (contd.) The Drift and Diffusion coefficients and are

calculated as follows :

The Fokker-Planck equation may be written as

where W is the probability density and the probability current is defined by

01

m

i i

i

C

S

t

W

)},({)},({)},({ tXgX

tXgtXhD ijk

kjii

)},({)},({, tXgtXgD ijijji

iD jiD ,

j j

ijii C

WDWDS

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:

The randomness only occurs in the linear part, the non-linear portion is deterministic.

Formally, the solution for step is

which can be approximated as

kk aNLLdz

dai )(

)()](exp[)( zaNLLzizza kk

z

)(2

expexp2

exp zazL

izNLizL

i k

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

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.

3)( z

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.

We can make the following approximations

d

LB

1d

LBan

),()( xzAzan

2

22

1 2),(),()(

x

Ad

x

AdxzAdxzAzan

Proposed Work – Continuum Approximation (contd.) Substituting for the coupling terms, we have

This reduces the governing equation to a 1D Nonlinear Schrodinger Equation

We will extend the continuum approximation to 2D

2

22

11 2)()(x

AdAzaza nn

Ax

x

AdCCAAA

dz

dAi

~

2

22

00

22

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