Linearized finite-element method solution of the ion-exchange nonlinear diffusion model

Mohamed M. Badr; Mohamed A. Swillam

doi:10.1117/1.JNP.11.026013

20 June 2017 Linearized finite-element method solution of the ion-exchange nonlinear diffusion model

Mohamed M. Badr, Mohamed A. Swillam

Author Affiliations +

Journal of Nanophotonics, Vol. 11, Issue 2, 026013 (June 2017). https://doi.org/10.1117/1.JNP.11.026013

Abstract

Ion-exchange process is one of the most common techniques used in glass waveguide fabrication. This has many advantages, such as low cost, ease of implementation, and simple equipment requirements. The technology is based on the substitution of some of the host ions in the glass (typically Na+) with other ions that possess different characteristics in terms of size and polarizability. The newly diffused ions produce a region with a relatively higher refractive index in which the light could be guided. A critical issue arises when it comes to designing such waveguides, which is carefully and precisely determining the resultant index profile. This task has been proven to be hideous as the process is generally governed by a nonlinear diffusion model with no direct general analytical solution. Furthermore, numerical solutions become unreliable—in terms of stability and mean squared error—in some cases, especially the K+−Na+ ion-exchanged waveguide, which is the best candidate to produce waveguides with refractive index differences compatible with those of the commercially available optical fibers. Linearized finite-element method formulations were used to provide a reliable tool that could solve the nonlinear diffusion model of the ion-exchange in both one- and two-dimensional spaces. Additionally, the annealed channel waveguide case has been studied. In all cases, unprecedented stability and minimum mean squared error could be achieved.

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Mohamed M. Badr and Mohamed A. Swillam "Linearized finite-element method solution of the ion-exchange nonlinear diffusion model," Journal of Nanophotonics 11(2), 026013 (20 June 2017). https://doi.org/10.1117/1.JNP.11.026013

Received: 1 March 2017; Accepted: 30 May 2017; Published: 20 June 2017

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