**4.2. The relationship between birefringence and stresses**

Quantification of changes in the refractive index of glass, in which the mechanical stresses arise, requires knowledge of the nature of the stress and elastooptic constants of the glass. The resulting stresses in the doped glass are of compressive nature [41]. In the case of a planar waveguide structure, for which the doped region extends from the surface into the glass, there is a one-dimensional distribution of the admixture concentration. In a direction perpendicular to the surface of the glass the doped region is free to deform, as a result of which at this direction

**Figure 14.** Determination of refractive birefringence and modal birefringence in the planar waveguide.

the stresses do not occur σxx(x) = 0. In contrast, at the directions parallel to the glass surface, the stresses generated in the doped area will be the same: σxy(x) = σxz(x). The geometries of the directions of the stresses are shown in Fig.15. These stresses are the functions of a depth *x* only, the same as one-dimensional concentration distribution of the admixture introduced into the glass.

**Figure 15.** The geometries of the directions of the stresses in a planar waveguide.

Changes in the refractive index of the glass, which result from the generated stress, are determined by the elastooptic constants [42]:

$$\mathbf{C}\_1 = \frac{d\mathbf{n}\_{\parallel}}{d\sigma} \quad \left(\frac{m^2}{N}\right) \qquad \mathbf{C}\_2 = \frac{d\mathbf{n}\_{\perp}}{d\sigma} \quad \left(\frac{m^2}{N}\right) \tag{17}$$

In the above equations, dn// and dn mean the differentials of change in refractive index for the wave with polarizations respectively parallel and perpendicular to the direction of stress σ.

The refractive index profiles nTM(x) and nTE(x) for the waveguides with TM and TE polariza‐ tions, propagating in the direction of the axis z (Fig.15), in a waveguide are (in the presence of stress) described by the Maxwell-Neumann equations [41,43]:

Producing the Gradient Changes in Glass Refraction by the Ion Exchange Method — Selected Aspects http://dx.doi.org/10.5772/60641 125

$$\begin{aligned} n\_{\rm TM}\left(\mathbf{x}\right) &= n\_0\left(\mathbf{x}\right) + \mathbf{C}\_1 \sigma\_{\rm xx}\left(\mathbf{x}\right) + \mathbf{C}\_2 \left[\sigma\_{yx}\left(\mathbf{x}\right) + \sigma\_{zx}\left(\mathbf{x}\right)\right] \\ n\_{\rm TE}\left(\mathbf{x}\right) &= n\_0\left(\mathbf{x}\right) + \mathbf{C}\_1 \sigma\_{yx}\left(\mathbf{x}\right) + \mathbf{C}\_2 \left[\sigma\_{zx}\left(\mathbf{x}\right) + \sigma\_{zx}\left(\mathbf{x}\right)\right], \end{aligned} \tag{18}$$

where n0(x) is a refractive index profile of the waveguide in the absence of stress.

After taking into account the assumptions: σxx(x) = 0 and σyx(x) = σzx(x) = σ(x), the equations (18) reduce to the following form:

$$\begin{aligned} n\_{\rm TM}\left(\mathbf{x}\right) &= n\_0\left(\mathbf{x}\right) + 2\mathbf{C}\_2\sigma\left(\mathbf{x}\right) \\ n\_{\rm TE}\left(\mathbf{x}\right) &= n\_0\left(\mathbf{x}\right) + \left(\mathbf{C}\_1 + \mathbf{C}\_2\right)\sigma\left(\mathbf{x}\right) \end{aligned} \tag{19}$$

These relationships enable us to align the birefringence of refraction (15) with the distribution of stress σ(x):

$$
\delta n\_{\sigma} \left( \mathbf{x} \right) = n\_{\text{TM}} \left( \mathbf{x} \right) - n\_{\text{TE}} \left( \mathbf{x} \right) = \left( \mathbf{C}\_2 - \mathbf{C}\_1 \right) \sigma \left( \mathbf{x} \right) \tag{20}
$$

The elastooptic constants for the BK-7 glass for the wavelength of *λ* = 677 nm are [40]:

the stresses do not occur σxx(x) = 0. In contrast, at the directions parallel to the glass surface, the stresses generated in the doped area will be the same: σxy(x) = σxz(x). The geometries of the directions of the stresses are shown in Fig.15. These stresses are the functions of a depth *x* only, the same as one-dimensional concentration distribution of the admixture introduced into the

**Figure 14.** Determination of refractive birefringence and modal birefringence in the planar waveguide.

Changes in the refractive index of the glass, which result from the generated stress, are

*dn m m dn C C dN dN*

^ æ ö æ ö = = ç ÷ ç ÷ ç ÷ ç ÷ è ø è ø

In the above equations, dn// and dn mean the differentials of change in refractive index for the wave with polarizations respectively parallel and perpendicular to the direction of stress σ.

The refractive index profiles nTM(x) and nTE(x) for the waveguides with TM and TE polariza‐ tions, propagating in the direction of the axis z (Fig.15), in a waveguide are (in the presence of

2 2

 s (17)

**Figure 15.** The geometries of the directions of the stresses in a planar waveguide.

/ / 1 2

s

stress) described by the Maxwell-Neumann equations [41,43]:

determined by the elastooptic constants [42]:

glass.

124 Ion Exchange - Studies and Applications

$$\mathbf{C\_1 = -0.5 \cdot 10^{-6} \quad \left(\text{mm}^2/\text{N}\right)}\qquad \mathbf{C\_2 = -3.3 \cdot 10^{-6} \quad \left(\text{mm}^2/\text{N}\right)}\tag{21}$$

They allow to specify the value of the stress generated in the doped area of the glass. As it stems from the equation (20):

$$\sigma\left(\mathbf{x}\right) = \frac{n\_{\rm TM}\left(\mathbf{x}\right) - n\_{\rm TE}\left(\mathbf{x}\right)}{\mathbf{C}\_2 - \mathbf{C}\_1} \tag{22}$$

The maximum stresses which occur at the surface of the BK-7 glass (*x* = 0), by doping it with potassium ions K+ and silver ions Ag+ , can be determined based on the data presented in Table 4. They are |σ(0)K+| = 678 (N/mm2 ) and |σ(0)Ag+| =375 (N/mm2 ) respectively.

Figure 16 illustrates the distributions of stresses occurring in the BK-7 glass after the diffusion processes of potassium ions K+ , and silver ions Ag+ , depending on their normalized concen‐ trations *u*. This presentation allows to compare these functions, defined on the basis of different refractive index profiles.

#### **4.3. The dependence of stress birefringence on the duration of the diffusion processes**

During the doping of the glass, the emerging stresses are also accompanied by the relaxation processes. They are reflected in the reduction of the role of stresses in the changes of the

**Figure 16.** The dependence of stresses on normalized concentration of admixture in the BK-7 glass.

refractive index of the doped area of the glass. This phenomenon can be observed by comparing the refractive index profiles of waveguides produced in processes with different diffusion times.

The tests were carried out on BK-7 glass doped with potassium ions K+ in the diffusion processes lasting from 24 to more than 500 h. The resulting waveguide structures were multimode. This guaranteed the fidelity of reconstruction of their refractive index profiles.

**Figure 17.** Refractive birefringence (at the glass surface) and modal birefringence, depending on the duration of the diffusion process.

The methodology of the experiment was as follows: 21 glass substrates made of BK-7 glass with the dimensions: 8 × 30 × 1.5 mm were prepared. They were polished on one side and then placed in the holders of silica glass tubes, allowing their individual extraction from the crucible containing the molten potassium nitrate. Due to the long duration of the process, the volume of salt (KNO3) filling the crucible was large (~300 cm3 ). During the entire process, the molten salt was continuously stirred. This ensured thermal and concentration uniformity for the entire bath. On alternate days an additional portion of salt was introduced into the crucible in order to supplement salt losses as a result of evaporation. From all the 21 substrates placed at the beginning of the process in the crucible, one was removed every 24 h. This substrate, after cooling and washing in distilled water, was subjected to further testing. The duration of the whole process was 504 h. The temperature of the salt in the crucible was measured with a thermocouple and recorded continuously. The averaged temperature was (399 ± 1)°C.

The measurements of synchronous angles of modes in produced waveguides were performed using a prism coupler [33] made of PSK-3 glass (of Schott company). They were carried out for a wavelength *λ* = 677 nm and the TE and TM polarization. The effective refractive indices of the modes Nm calculated on their basis have the uncertainty of measurement at the level of ∆Nm = 0.0002.

Figure 17 shows the change in the refractive birefringence δnσ(0) at the surface of the wave‐ guides and in modal birefringence δNσ(m) for the modes of 0-6 row, depending on the duration of the diffusion process. The course of the changes δnσ(0) indicates a decrease in the birefrin‐ gence on the surface of the waveguide with increasing duration of the diffusion. In contrast, the courses of modal birefringence tend to increase with the diffusion time. However, the nature of this growth is a function of the order of the mode. In the case of a zero-order mode, even a downward trend can be seen. The strongest increase in birefringence occurs for the modes of higher-order. The modal birefringence (16) is not related to the location in the waveguide, where there is a defined concentration of the admixture. It can be however assumed, that it reflects the nature of the stresses associated with the depth of diffusion of admixture into the glass.
