Solving the nonlinear diffusion model of the ion exchange process using finite element method

Abstract

Due to its low cost and simplicity, ion exchange is considered one of the most commonly used processes to produce glass waveguides nowadays. This fabrication technology is based on the substitution of some ions already present in the glass with other ions having different sizes and polarizabilities. A careful study of the resultant refractive index profile is crucial in the impact on the waveguide characteristics. In this paper, we introduce, for the first time, a novel solution of the nonlinear diffusion equation that model this process using finite element method (FEM) approach. The ion exchange can be modelled as a nonlinear diffusion equation, as the exchanged ions Ka+ diffuse into their new sites where the original ions were existing. Numerical instabilities are encountered when solving for the exchanged ions with similar diffusion coefficients as in the case of Ka+/Na+, which is used in fabricating integrated optical devices with refractive index differences compatible with those of the optical fibers. Different novel FEM techniques are proposed in solving the problem in 1D space. The stability and accuracy of the different methods outperforms the current numerical methods and provide a good tool for highly nonlinear diffusion problem.