## 2 Theoretical Background  
Changes in the refractive index in glasses, which are the effect of the resulting stresses, are determined by means of elasto-optic constants [16]:  
\[C_{1}=\frac{dn_{//}}{d\sigma}\left(\mathrm{Pa}^{-1}\right),\ C_{2}=\frac{dn_{ \perp}}{d\sigma}\left(\mathrm{Pa}^{-1}\right) \tag{1}\]  
In the above equations, \(\mathrm{dn}_{//}\) and \(\mathrm{dn}_{\perp}\) mean differentials of change in the refractive index for a wave with polarizations correspondingly parallel and perpendicular to the direction of stress \(\sigma\). The elasto-optic constants thus defined can also be expressed by the following compounds [16]:  
\[ \begin{array}{c}C_{1}=-\frac{n_{0}^{3}}{2E}(p_{11}-2\mu p_{12})\\ C_{2}=-\frac{n_{0}^{3}}{2E}[-\mu p_{11}+(1-\mu)p_{12}],\end{array}  \tag{2}\]  
where:  
\(n_{0}\)--refractive index of the glass in the absence of stress,  
\(p_{11}\), \(p_{12}\)--elasto-optic coefficients,  
\(E\)--Young's modulus,  
\(\mu\)--Poisson's ratio. The knowledge of the elasto-optic coefficients \(p_{11}\) and \(p_{12}\), the Young's modulus and the Poisson's ratio [16] allows, based on (2), to calculate the elasto-optic constants for the BK7 glass. These values for the wavelength \(\lambda\) = 677 nm are [17]:  
\[C_{1}=-0.5\times 10^{-6}\left(\mathrm{MPa}^{-1}\right),\ C_{2}=-3.3\times 10^{-6} \left(\mathrm{MPa}^{-1}\right) \tag{3}\]  
The processes of ion exchange in glass with the use of a liquid source of admixture ions, among which the most widely used are nitrates, are carried out at temperatures much lower than the transition temperature of glasses \(T_{g}\). For the borosilicate glass BK-7, this temperature \(T_{g}\) = 557 \({}^{\circ}\)C [16] is much higher than the temperature \(T_{diff}\)-400 \({}^{\circ}\)C of the implementation of the K\({}^{+}\)\(\leftrightarrow\)Na\({}^{+}\) ion exchange processes with the use of liquid potassium nitrate KNO\({}_{3}\) as the source of K\({}^{+}\) ions. In such cases, changing the refractive index of the glass in its surface area where the ion exchange occurs is due to not only the difference of their electric polarizability, but also as the result of the elasto-optic phenomenon generated by mechanical stresses occurring in this area [13]. These stresses are the result of changes in the volume of glass in the doping area, which results from the difference of ionic radii of exchanged components, as well as the difference in thermal expansion between the doped region and the rest of the glass. The geometry of these stresses is shown in Figure 1. The presence of these stresses in the waveguide layer of the glass is revealed during the propagation of the electromagnetic wave in it. Then there is a difference between the constant propagation of the modes of the same order for a monochromatic wave, depending on the state of its polarization: TE or TM. The refractive index profiles corresponding to these polarization states are described in the following equations [13]:  
\[ \begin{split} n_{TM}(x)&=n_{0}(x)+C_{1}\sigma_{xx}(x)+C _{2}\big{[}\sigma_{yx}(x)+\sigma_{zx}(x)\big{]}\\ n_{TE}(x)&=n_{0}(x)+C_{1}\sigma_{yx}(x)+C_{2}[ \sigma_{xx}(x)+\sigma_{zx}(x)],\end{split}  \tag{4}\]  
where: \(n_{TM}(x)\), \(n_{TE}(x)\)--refractive index profiles for TM and TE polarizations, \(n_{0}(x)\)--refractive index profile of a waveguide in the absence of stresses. After taking into account the assumptions: \(\sigma_{xx}(x)=0\) and \(\sigma_{yx}(x)=\sigma_{zx}(x)=\sigma(x)\), the Equation (4) simplify to the form:  
\[ \begin{split} n_{TM}(x)&=n_{0}(x)+2C_{2}\sigma(x) \\ n_{TE}(x)&=n_{0}(x)+(C_{1}+C_{2})\sigma(x)\end{split}  \tag{5}\]  
They allow to determine the value of stresses created in the doping area of the glass. From the Equation (5) results the following:  
\[\sigma(x)=\frac{n_{TM}(x)\text{-}n_{TE}(x)}{C_{2}\text{-}C_{1}} \tag{6}\]  
The refractive index profiles of a waveguide for TE and TM polarization and the refractive index profile in the absence of stresses can be represented in the form [14]:  
\[