1. 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

2. Outline
Motivation
The Optics Model
Proposed Work

3. 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.

4. The Optics Problem
2-D Hexagonal Optical Fiber Array

5. Governing Equation Consider the nonlinear wave equation (*)
The solution is given by
Ignoring the non-linearity and substituting into the wave equation (*), we have

6. Governing Equation (contd.)
We substitute the following representations into equation (**)
which gives

7. 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.

8. 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

9. 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.

10. Numerical Solution (contd.)

11. 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.

12. 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.

13. 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

14. 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.

15. 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

16. 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.

17. Numerical Solution of the SDE (contd.)

18. 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.

19. Numerical Solution of the SDE (contd.)

20. Numerical Solution of the SDE (contd.)

21. 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 an LB

22. 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

23. 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