# Integrated Photonics on Glass: A Review of the Ion-Exchange Technology Achivements
## 2 Ion-Exchanged Waveguides
### Modelling Ion-Exchanged Waveguides  
Extensive work has been carried-out throughout the years to characterize and model ion-exchanges processes [18; 19; 20; 21; 22]. In this article, we will focus on a relatively simple description since it occurred to be reliable enough to allow us designing waveguides and predicting their optical behavior efficiently. Ion-exchange can be seen as a two-step process: first the exchange itself that occurs at the surface of the glass and creates a normalized  
Figure 4: Realization of deeply buried waveguides by an Ag\({}^{+}\)/Na\({}^{+}\) ion-exchange on a GO14 Teem-Photonics glass. The applied electric field during the burying process was 650 kV/m; (**a**) image of the output of the waveguide observed with an InGaAs Camera at \(\lambda\) = 1.5 \(\upmu\)m; (**b**) vertical cut of the measured intensity showing the position of the mode with respect to the glass water substrate. Figure 3: Image of a quasi-circular waveguide observed with an optical microscope, the glass is in light blue, the core is in pink, air is in dark blue. Figure 2: (**a**) Thermal burying of a waveguide’s core; (**b**) electrically assisted burying of the waveguide’s core. The competition between ionic diffusion and transport allows obtaining quasi circular profiles. concentration \(c_{s}\) of doping ions. For thin film sources, this concentration is linked to the applied current by the following relation:  
\[\frac{\partial c_{s}}{\partial x}=\frac{J_{0}(c_{s}-1)}{D_{Ag}}, \tag{3}\]  
where \(J_{0}\) is the ion flux created by the electrolysis, \(x\) is the direction normal to the surface, and \(D_{Ag}\) is the diffusion coefficient of silver in the glass. For liquid sources made of a mixture of molten salts containing B\({}^{+}\) and A\({}^{+}\) ions in order to replace A\({}^{+}\) ions of the glass, an equilibrium at the glass surface is usually rapidly reached, according to the chemical reaction:  
\[A_{salt}^{+}+B_{glass}^{+}\Leftrightarrow B_{salt}^{+}+A_{glass^{\prime}}^{+} \tag{4}\]  
Considering that the amount of ions in the molten salt is much bigger than the one of the glass, the ion concentrations in the liquid source can be considered as constant, which allows deriving the relative concentration at the surface:  
\[c_{s}=\frac{Kx_{B}}{1+x_{B}(K-1)}, \tag{5}\]  
\(K\) being the equilibrium constant of the chemical reaction (4) and \(x_{B}=C_{B}^{salt}/\left(C_{B}^{salt}+C_{A}^{salt}\right)\) is the molar fraction of doping ions B\({}^{+}\) in the molten salt. Since the refractive index is proportional to the relative concentration, according to (2), it is easy to fix the refractive index change at the glass surface by setting the ratio of B\({}^{+}\) ions in the liquid source. Figure 5 shows an experimental determination of this dependence for a silver/sodium ion-exchange on a Schott-BF33 glass. These data have been obtained by realizing highly multimode slab waveguides and retrieving their refractive index profile through m-lines measurements [23] and the Inv-WKB procedure [24; 25]. The ions exchanged at the glass surface entail a gradient of concentration inside the glass. Hence, B\({}^{+}\) ions migrate inside the glass while A\({}^{+}\) ions are moving towards the surface. Since the two species of ions have different mobilities, an internal electrical field \(\overrightarrow{E_{int}}\) is created during the diffusion process. To this field an external applied field \(\overrightarrow{E_{app}}\) can  
Figure 5: Refractive index change measured at the surface of a Schott-BF33 glass for different \(x_{B}\)AgNO\({}_{3}+(1-x_{B})\)NaNO\({}_{3}\) molten salts at a temperature of 353 \({}^{\circ}\)C.  
be added, which results in ions fluxes \(\overrightarrow{J_{A}}\) and \(\overrightarrow{J_{B}}\), for A\({}^{+}\) and B\({}^{+}\), respectively, which are determined by the Nernst-Einstein equation:  
\[