Electro-diffusion equations of monovalent cations in glass under charge neutrality approximation for optical waveguide fabrication

Two improvements of the usual three-dimensional partial differential equations that model ion-exchanged optical devices in glass under charge neutrality approximation are proposed. First, they are rigorously generalised for both non-ideal cation behaviour and concentration-dependent self-diffusion coefficients. Secondly, the Faraday equation is imposed in the derivation instead of the Ohm law. Accordingly, the current density distribution is not governed by the electric field but by the gradient of an effective electrical potential instead. On the other hand, the boundary conditions of those equations are obtained from standard electrolyte theory, which results in contact potentials at interfaces between glass and molten salt, silver film or metallic cathode. Therefore, typical arrangements for field-assisted ion exchange must be considered as electrochemical cells, and thus a correction term must be added to the applied voltage to get a proper modelling. This term can be neglected in some typical cases but it must be retained in others. Finally, it is predicted that the electrochemical cell effect can be used to diffuse cations from a silver film without need for an external applied field. This suggests a new simple method to fabricate integrated optical components by ion exchange in glass.