A General Analytical Solution for the Concentration Profiles of Ion-Exchanged Planar Waveguides

Abstract

ABSTRACTDuring the recent years, a great variety of ion-exchange processes, including one-step or two-step electric field assisted ion-exchange processes, have been developed to fabricate different kinds of passive planar glass waveguides, e.g., surface waveguides, which correspond to surface maximum concentration, or buried waveguides, which correspond to inside maximum concentration [1,2,3]. Theoretical calculation of ionic concentration distribution has been of interest since refractive index is generally a linear function of concentration. A general analytical solution to calculate both surface and buried concentration distributions from different ion-exchange processes, however, has not yet been presented. In addition, traditional ion-exchange has been carried out only with constant surface concentration boundary conditions. Little attention has been paid, either experimentally or theoretically, to ion-exchange processes with variable boundary conditions. In fact, the time-dependent surface concentration is experimentally observed for the ion-exchange of GRIN glass in molten salt bath [4]. Very recently, a novel one-step technique [5,6] involving electric field assisted ion-exchange of Na+ in glass by Ag+ from molten AgNO3 bath with decaying silver concentration has been developed to produce buried Ag+ concentration profiles in glass. As the accurate and reproducible processes are very important for fabricating ion-exchanged glass waveguides, theoretical modeling and analysis on the new process are needed.In this paper, the one-dimensional field-assisted linear diffusion equation has been analytically solved by Laplace transformation to theoretically calculate concentration profiles produced by field enhanced ion-exchange process with exponentially decaying surface concentration boundary conditions. The applications of the solution to a variety of ion-exchange processes with different boundary or processing conditions for optical waveguide fabrication have been discussed. The theoretical results prove that the solution is a general analytical solution which can be used to calculate either surface concentration profiles or buried concentration profiles.