## 2 Fundamentals of Ion-Exchanged Glass Waveguide Circuits
The centers of the channel guides (usually the position of the maximum refractive index) follow paths described by functions \(f_{j}(z)\), that is, \(z\)-functions representing translations of a refractive index profile. A typical case is a channel guide \(j\), which is bent along S-shaped transitions to couple it with other parallel channel guide \(j+1\). These transitions may follow, for example, the path described by [PERSON] et al. (Minford curve) for obtaining low losses [62]. An example of a refractive index profile representation for an arbitrary channel guide \(j\) is the following (from here and for the sake of simplicity we omit subscript \(j\))  
\[\Delta n(x,y,z)=\Delta n\operatorname{Erfc}\bigl{(}\frac{x}{d_{ox}}\bigr{)}\, \exp\bigl{\{}-\frac{(y-f(z))^{2}}{d_{oy}^{2}}\bigr{\}} \tag{2}\]  
where Erfc is the complementary error function, which is a typical solution for linear ion exchange, although more general solutions can be chosen [63; 64; 65; 17], \(f(z)\) is the curve (path) profile, \(\Delta n\) is the maximum refractive index change and \(d_{ox}\) and \(d_{oy}\) are effective depths along \(x\) and \(y\) direction. Note that we have assumed a factorizable index profile, \(\Delta n(x,y,z)=\Delta n\,E(x)G(y,z)\), that is, a product between an index profile in depth (x-direction) and a Gaussian index profile along \(y\) and \(z\) directions describing quite well the lateral diffusion. In many cases such a factorization is a good approximation, or even it can be forced for the sake of design simplicity [66]. Moreover, it is well known that there are different approximate methods to be used in order to obtain manageable mode solutions such as, for example, the effective index method (EIM) [66; 16], perturbative and variational methods [16] and so on. On the other hand, this kind of function can also describe the change of index or geometry for other integrated optical elements as phase shifters, for  
Figure 6: A general device formed with \(N\) paths (channel waveguides) in active (**left**) and passive (**right**) sections of a hybrid glass substrate. An optical grating (OG) in the rare-earth-doped glass section (the left part of the substrate) is used to implement a laser structure which, pumped by beams \(\mathrm{P}_{j}\), \((j=1,\ldots,N)\), generates \(N\) laser signals; several directional couplers (DC) and phase shifters (PS) in the passive section perform linear optical transformations. \(N\) detectors \(\mathrm{D}_{j}\) are also shown. example, those based on an adiabatic approximation that could be represented by the following function  
\[\Delta n(x,y,z)=\Delta n\operatorname{Erfc}(x/d_{ox})\,\exp\{-\frac{y^{2}}{d_{y}^{ 2}(z)}\} \tag{3}\]  
Note that in this case we have a rectilinear channel waveguide (\(f(z)=0\)) with \(d_{y}(z)\) an adiabatic \(z\)-function and therefore the effective index changes slowly on \(z\) and thus a well defined phase shifting can be obtained. We must recall that the cornerstone of the ion-exchange method is that these functions are, in a good approximation, proportional to the concentration \(C_{j}(x,y,z)\) of exchanged ions. In fact, the \(q\) parameters \(p_{1},\ldots,p_{q}\) such as maximum refractive index change \(\Delta n\), effective depth \(d_{ox}\), lateral effective depth \(d_{oy}\) and so on, can be expressed in an empiric way as functions of both the fabrication physical parameters, such as temperature \(T\), ionic concentration \([C]\) of the molten salt (or ionic concentrations \([C_{1},C_{2},\ldots]\), as in the case of molten salts eutectic mixtures), ion-exchange time \(t\) and also geometrical parameters, as width of the channel guide mask \(w\), separation between channel guides \(s\) and so on.