**4.1. Refractive birefringence and modal birefringence**

change in volume in the case of K+ ↔ Na+

of thermal expansion of glass is:

122 Ion Exchange - Studies and Applications

component in glass respectively.

and *ci*

where *α<sup>i</sup>*

**Type of glass**

soda-lime nb=1.5090

BK-7 nb=1.5137

(data provided by refraction profiles of Fig.13).

exchange is much larger than in the Ag+ ↔ Na+

= × å *<sup>c</sup>* (14)

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

, *i i*

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.

represent, the coefficient of thermal expansion and the mole fraction of i-th

**Type of admixture ions Ag+ K+**

ns,TE=1.6049 ∆ns,TE=0.0959 ns,TE=1.5163 ∆ns,TE=0.0073 ns,TM=1.6062 ∆ns,TM=0.0972 ns,TM=1.5176 ∆ns,TM=0.0086

ns,TE=1.5929 ∆ns,TE=0.0792 ns,TE=1.5219 ∆ns,TE=0.0082 ns,TM=1.5939 ∆ns,TM=0.0802 ns,TM=1.5238 ∆ns,TM=0.0101

**Table 4.** Changes in the refractive index (for *λ* = 677nm) at the glass surface depending on the type of admixture ions

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

*i*

 a

a

In the description of the phenomenon of birefringence, which extends over the entire doping area of glass, we can use the concept of birefringence as a function whose domain is the depth of doping area of the glass, or the row of modes. The following definitions of stress birefrin‐ gence refractive δnσ and modal δNσ are introduced.

Refractive birefringence is determined by the refractive index profiles for the TE and TM polarization of the waveguide structure and is a function of the depth *x* counted from the glass surface:

$$
\delta n\_{\sigma} \left( \mathbf{x} \right) = n\_{\text{TM}} \left( \mathbf{x} \right) - n\_{\text{TE}} \left( \mathbf{x} \right) \tag{15}
$$

The modal birefringence concerns the difference in the effective refractive indices of the modes of the same order for the TE and TM polarization. This value is a function of the mode order:

$$
\delta \mathcal{N}\_{\sigma} \left( m \right) = \mathcal{N}\_{\text{TM}} \left( m \right) - \mathcal{N}\_{\text{TE}} \left( m \right) \tag{16}
$$

The sense of the values incorporated herein is illustrated in Fig.14. Refractive birefringence describes the difference in the refractive indices of the refractive index profiles for the TE and TM polarization that occur at a depth *x* in the waveguide. It is related to the stresses arising in the glass, in relation to the concentration of admixture introduced into the volume of the glass during the ion exchange processes. Function δnσ(x) reaches a maximum at the glass surface (*x* = 0) and decreases monotonically to zero at the point of glass, which was not reached by the admixture. The modal birefringence can be determined only for those rows of modes for which simultaneously there exist modes of both TE and TM polarization.
