Theoretical Aspects of the Ion Exchange in the Fabrication of Focusing Gradient-Index Lenses

An expression relating the refractive index at different points of the gradient region of an axially symmetric ion-exchanged gradient-index element to the concentration of the component incorporated into the glass from the salt melt is derived and analyzed. It is demonstrated that the radial distribution of the refractive index gradient in a focusing gradient-index element is described by an even function symmetric about the ordinate axis with a local minimum on it and two side maxima in the surface region of the cylinder. This provides an explanation for the origin of the surface “tails.” A theoretical relationship is obtained using the metric of the equilibrium chemical diagram. The correctness of this relationship is verified for the case of the exchange between lithium ions from an alkali borosilicate glass and sodium ions from the salt melt, as well as for the case of the exchange between thallium ions from a thallium borosilicate glass and potassium ions from the salt melt.