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 (*) The solution is given by Ignoring the non-linearity and substituting into the wave equation (*), we have

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

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.

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

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.

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.

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

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

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.

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 an LB

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

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