# Integrated Photonics on Glass: A Review of the Ion-Exchange Technology Achivements
## 2 Ion-Exchanged Waveguides
### Modelling Ion-Exchanged Waveguides
\begin{split}\overrightarrow{J_{A}}&=-D_{A}\left[ \overrightarrow{\
abla}C_{A}-\frac{e}{Hk_{B}T}C_{A}\left(\overrightarrow{E_ {int}}+\overrightarrow{E_{app}}\right)\right]\\ \overrightarrow{J_{B}}&=-D_{B}\left[\overrightarrow{ \
abla}C_{B}-\frac{e}{Hk_{B}T}C_{B}\left(\overrightarrow{E_{int}}+ \overrightarrow{E_{app}}\right)\right]\end{split}  \tag{6}\]  
where \(D_{i}\) is the diffusion coefficient of the ion i, \(C_{i}\) its concentration, \(e\) is the electron charge, \(k_{B}\) the Boltzmann constant, \(T\) the temperature and \(H\) the Haven coefficient. Assuming that all the sites left by ions A\({}^{+}\) are filled by ions B\({}^{+}\), it can be written that at any position in the glass the relation \(C_{A}+C_{B}=C_{A0}\), where \(C_{A0}\) is the concentration of A\({}^{+}\) ions before the exchange, is always valid. With this relation and Equation (6), the total ionic flux can be expressed as:  
\[ \begin{split}\overrightarrow{J}=\overrightarrow{J_{A}}+ \overrightarrow{J_{B}}&=-D_{A}C_{A0}\left[\alpha\overrightarrow{ \
abla}c-\frac{e}{Hk_{B}T}(1-\alpha c)\left(\overrightarrow{E_{int}}+ \overrightarrow{E_{app}}\right)\right],\end{split}  \tag{7}\]  
where the Steward coefficient \(\alpha=1-D_{B}/D_{A}\) and the normalized concentration \(c=C_{B}/C_{A0}\) have been introduced. If no electric field is applied, then the total current is null, which allows determining easily \(\overrightarrow{E_{int}}\):  
\[ \begin{split}\overrightarrow{E_{int}}=-\frac{Hk_{B}T}{e}\frac{ \alpha\overrightarrow{\
abla}c}{1-\alpha c}\end{split}  \tag{8}\]  
The second Fick's law implies that:  
\[ \begin{split}\frac{\partial C_{B}}{\partial t}=-\
abla \overrightarrow{J_{B}}.\end{split}  \tag{9}\]  
Combining (6), (8) and (9), the equation that governs the evolution of the relative concentration as a function of time is obtained:  
\[ \begin{split}\frac{\partial c}{\partial t}=\overrightarrow{ \
abla}\left[\frac{D_{B}}{1-\alpha c}\overrightarrow{\
abla}c-\frac{eD_{B}} {Hk_{B}T}c\overrightarrow{E_{app}}\right].\end{split}  \tag{10}\]  
Equation (10) can be solved numerically by Finite Difference or Finite Element schemes but for accurate modelling, the dependence of ionic mobility and diffusion on the concentration should not be neglected. The so-called mixed alkali effect plays indeed a significant role in ion-exchanges where a high doping concentration is required [26; 27]. It must also be noticed that ion-exchange modifies the conductivity of the glass, which in turn, modifies the field distribution of \(\overrightarrow{E_{app}}\). Therefore, solving Equation (10) is actually much less obvious than it might appear and handling these problems has been the subject of a quite abundant literature [28; 29; 30; 31]. Figure 6 displays typical refractive index profiles that have been obtained considering mixed alkali effect and the coupling between the ion-exchange and the applied electric field. Simulations have been done with an in-house software based on a finite difference scheme. It can be clearly seen how a proper choice of the experimental parameters can lead to circular waveguides. However, the maximum refractive index change is dropping from almost 0.1 to 10\({}^{-2}\) during the burial process because of the spreading of doping ions caused by thermal diffusion.