**4. Optical birefringence phenomenon of the gradient area as the effect of ion exchange**

Ion exchange in glass processes with the use of liquid source of admixture ions, among which the most widely used are nitrates, are carried out at temperatures much lower than transition temperature of glasses Tg. For example, for the borosilicate glass BK-7 (Schott), which is widely used in these processes, the temperature Tg = 557°C [40] is much higher than the temperature

Tdiff = 400°C of the implementation of the K+ ↔ Na+ ion exchange processes with the use of liquid potassium nitrate KNO3 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 not only to the difference of their electric polarizability, but also the result of the elastooptic phenomenon generated by mechanical stresses resulting in this area [41]. 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 components exchanged as well as the difference in thermal expansion between the doped region and the rest of the glass. the doping area, which results from the difference of ionic radii of components exchanged as well

as the difference in thermal expansion between the doped region and the rest of the glass.

Chemical composition in % by weight of the glass [38]: 72.2% SiO2, 14.3% Na2O, 6.4% CaO, 4.3% MgO, 1.2% Al2O3, 1.2% K2O, 0.3% SO3, 0.03% Fe2O3. Diffusion processes were carried out with a liquid source of admixture. The silver nitrate AgNO3 and its sodium nitrate NaNO3 solutions have been used. These solutions are determined by the molar fraction κ [39] listed

Figure 12 shows the dependence of the mass increase of the glass substrates from the product of the total surface of the glass and the integral of the normalized concentration of the

**Figure 12.** The dependence of the mass increase of the sample on the product of its surface and the integral of the nor‐

This dependency is based on the results shown in Table 3. For : MAg = 107.87 (g/mol), MNa = 22.99 (g/mol), NA = 6.02⋅1023 (mol-1), the following equation is obtained: A/c0 = 1.41⋅10-22 (g). Determined by this method the value of the equilibrium concentration in the glass is: *c*0 = A/

The results of the equilibrium concentration *c*0 obtained by integrating the electric charge (Section 2.1) and by method of weighing presented here, comply within the limits of uncer‐

**4. Optical birefringence phenomenon of the gradient area as the effect of**

Ion exchange in glass processes with the use of liquid source of admixture ions, among which the most widely used are nitrates, are carried out at temperatures much lower than transition temperature of glasses Tg. For example, for the borosilicate glass BK-7 (Schott), which is widely used in these processes, the temperature Tg = 557°C [40] is much higher than the temperature

in Table 3.

admixture (Ag+

120 Ion Exchange - Studies and Applications

ions).

malized concentration of the admixture.

1.41×10-22 = (5.6 ± 0.2) 1027(m-3).

tainty calculation.

**ion exchange**

Fig.13 Comparison of refractive index profiles of planar waveguides produced in soda‐lime glass doped with ions: Ag+ (a) or K+ (b) with the refractive index profiles produced in BK‐7 glass doped **Figure 13.** Comparison of refractive index profiles of planar waveguides produced in soda-lime glass doped with ions: Ag+ (a) or K+ (b) with the refractive index profiles produced in BK-7 glass doped with ions: Ag+ (c) or K+ (d).

with ions: Ag+ (c) or K+ (d). The ionic radii of the most commonly used admixtures K+ and Ag<sup>+</sup> are 1.33 Å and 1.26 Å The ionic radii of the most commonly used admixtures K+ and Ag+ are 1.33 Å and 1.26 Å [41] respectively. Thus, in relation to the ionic radius of sodium (0.95 Å [41]), which, due to one of the lowest binding energies with glass network, is its most easily replaceable component, the

[41] respectively. Thus, in relation to the ionic radius of sodium (0.95 Å [41]), which, due to one of the lowest binding energies with glass network, is its most easily replaceable component, the

exchange. After completion of the process, when the glass is cooled to a low room temperature, a resulted difference generates stress in the doping area of the glass. Also, the difference in thermal expansion of the doping area, in relation to the rest of the glass, makes a significant contribution to

exchange is much larger than in the Ag<sup>+</sup>

Na<sup>+</sup>

Na<sup>+</sup>

change in volume in the case of K+

change in volume in the case of K+ ↔ Na+ exchange is much larger than in the Ag+ ↔ Na+ exchange. After completion of the process, when the glass is cooled to a low room temperature, a resulted difference generates stress in the doping area of the glass. Also, the difference in thermal expansion of the doping area, in relation to the rest of the glass, makes a significant contribution to the resulting stress. According to the principle of additivity [41], the coefficient of thermal expansion of glass is:

$$\alpha = \sum\_{i} \alpha\_{i} \cdot c\_{i} \, \prime \tag{14}$$

where *α<sup>i</sup>* and *ci* represent, the coefficient of thermal expansion and the mole fraction of i-th component in glass respectively.

The coefficients of thermal expansion for sodium and potassium are αNa = 39.5⋅10-6 K-1 and α<sup>K</sup> = 46.5⋅10-6 K-1 [41] respectively. These values are related to the concentration of Na2O and K2O in the glass. These data indicate that ion exchange processes of K+ ↔ Na+ in the glasses will be accompanied by the generation of significant mechanical stresses. The presence of these stresses in the waveguide layer of the glass is apparent in the propaga‐ tion of the electromagnetic wave. A difference in propagation constants of the modes of the same order occurs for a monochromatic wave depending on its state of polarization. This phenomenon is called stress birefringence. Its scale depends on the type of glass and the admixture into the glass. This is illustrated in Fig.13, in which the refractive index pro‐ files of planar waveguides produced in two types of glass (soda-lime and BK-7) by doping them with silver ions Ag+ and potassium ions K+ are presented. These profiles were determined by measuring the propagation constants for the two polarization states: TE and TM. Measurement uncertainties of the effective refractive indices of the modes do not exceed 3⋅10-4. In all the presented refractive index profiles, the modes with TM polariza‐ tion have bigger values of effective refractive indices. As there is only a polarization mode dispersion it is therefore inversely than in the case of a waveguide without stresses.


**Table 4.** Changes in the refractive index (for *λ* = 677nm) at the glass surface depending on the type of admixture ions (data provided by refraction profiles of Fig.13).

For each case of the refractive index profile shown in Fig.13, the relative change in the refractive index at the glass surface for both polarization states, in relation to the TE polarization, has been shown. These figures can provide a quantitative measure of the scale of the birefringence when comparing this phenomenon in various glass-admixture systems. A comparison of Fig. 13a,c shows that the size of the birefringence due to the doping of both types of glasses with the silver ions Ag+ is practically the same, although the changes in refractive indices at the surface differ quite significantly (Table 4). In the case of doping both kinds of glass with potassium ions K+ the measure of the birefringence defined above is significantly higher in the case of BK-7 glass.
