**4.4. The changes of stress birefringence in the heating processes**

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

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

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

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

times.

126 Ion Exchange - Studies and Applications

diffusion process.

The relaxation phenomenon of a waveguide layer, in which the doping of the glass resulted in the appearance of stresses, is clearly visible after the heating processes. In order to observe this phenomenon, in the BK-7 glass the long-term heating processes were implemented in relation to the waveguide produced by the diffusion process of KNO3 in 72 hours [44]. The durations of these heating processes, after which the measurements of effective refractive indices of modes for both polarization states made were 24, 48, 96, 192, and 384 h respectively. The temperature of the heating processes was (399 ± 1)°C.

Based on the determined effective refractive indices, the refractive index profiles of the waveguides were reconstructed and their modal birefringence as well as the refractive birefringence at the surface of the glass was estimated. The nature of the changes of these values depending on the duration of the heating process is shown in Fig.18a.

Birefringences for the time *t* = 0 shown in the graphs represent the values of these terms after the diffusion processes. In the case of the refractive birefringence at the glass surface, the character of its changes is always a decreasing function of the duration of the heating process. Whereas, for the modal birefringence, the course of the dependence of δNσ(m) from the

processes (b).

duration of the heating is dependent on the order of *m* mode, which can be clearly seen in the cases of the modes of higher order. For the modes of 5th and 6th order (Fig.18a), the increase of the heating duration in the first 25 h is accompanied by the increase of the birefringence. This effect can be explained by the generation of stresses in the deeper areas of the glass, which are reached by the admixture due to the diffusion widening of the doping area during the heating of the waveguide structure.

On the basis of equation (22) and elastooptic constants (21) it can be calculated how the stresses at the glass surface change as a result of the heating processes. A graph of this relationship is shown in Fig.18 b. There can be seen that after a heating time of about 50 h, the value of stresses at the glass surface decreases to half its value after the diffusion process.

Fig.18 Refractive birefringence (at the glass surface) and modal birefringence depending on the time of the heating process Theat=(399 ± 1)°C (a). Changes in stresses at the glass surface in the heating **Figure 18.** Refractive birefringence (at the glass surface) and modal birefringence depending on the time of the heating process Theat=(399 ± 1)°C (a). Changes in stresses at the glass surface in the heating processes (b).

#### **4. Repeatability of the technological effects of diffusion manufacturing 5. Repeatability of the technological effects of diffusion manufacturing processes of gradient changes in the refractive index of glasses**

**processes of gradient changes in the refractive index of glasses**  The ion exchange in glass method can be considered as a technological method. With its use, the predictable and repeatable results are obtained. Those effects are the changes in glass refraction, which are of a gradient character. They are described by the refractive index profiles. Determination of the refractive index profiles is easy to implement in the case of planar waveguides. For this purpose a waveguide method with the use of selective excitation of modes using a prism coupler is used [33]. In this method, the measurements of synchronous angles are used, for which excitations of waveguide modes are obtained. Such measurements are made using The ion exchange in glass method can be considered as a technological method. With its use, the predictable and repeatable results are obtained. Those effects are the changes in glass refraction, which are of a gradient character. They are described by the refractive index profiles. Determination of the refractive index profiles is easy to implement in the case of planar waveguides. For this purpose a waveguide method with the use of selective excitation of modes using a prism coupler is used [33]. In this method, the measurements of synchronous angles are used, for which excitations of waveguide modes are obtained. Such measurements are made using a goniometer. Precision of the measurements of these angles is at the level of several tens of seconds. Based on the measured synchronous angles, the effective refractive

a goniometer. Precision of the measurements of these angles is at the level of several tens of seconds. Based on the measured synchronous angles, the effective refractive indices of the waveguide modes are calculated. The uncertainties of their determination are at the level of 10-4. Based on the set of effective refractive indices of modes, a reconstruction of the refractive index profile of the waveguide is performed. For this purpose the procedure proposed in 1976 by White and Heidrich [45] is used. This procedure is based on the mode equation of the waveguide. Its

The control of technological processes of diffusion doping of glasses in the ion exchange

processes was proposed in the work [33] (p.181). The described method is used in producing one-

specific application is shown in [33] (p.169).

indices of the waveguide modes are calculated. The uncertainties of their determination are at the level of 10-4. Based on the set of effective refractive indices of modes, a reconstruction of the refractive index profile of the waveguide is performed. For this purpose the procedure proposed in 1976 by White and Heidrich [45] is used. This procedure is based on the mode equation of the waveguide. Its specific application is shown in [33] (p.169).

The control of technological processes of diffusion doping of glasses in the ion exchange processes was proposed in the work [33] (p.181). The described method is used in producing one-dimensional planar waveguides. The basis of this method is to determine the temperature dependences of diffusion coefficients of ions exchanged in a glass-admixture system. Knowl‐ edge of these relationships allows to calculate the kinetics of diffusion process in the current process temperature. This is done by measuring the temperature of the process carried out by a thermocouple placed in the immediate vicinity of the glass plate. This measurement occurs at defined points in time tp. Moments of the temperature measurement are a total multiples of ∆t time step which is numerically integrated over one-dimensional diffusion equation [33] (p. 177):

$$\frac{\partial \mathbf{\hat{u}}}{\partial \mathbf{t}} = \frac{\mathbf{D}\_{\alpha \mathbf{k}} \mathbf{e}^{\mathbf{A} \mathbf{u}}}{1 - a \mathbf{u}} \cdot \frac{\partial^2 \mathbf{u}}{\partial \mathbf{x}^2} + \frac{\mathbf{D}\_{\alpha \mathbf{k}} \mathbf{e}^{\mathbf{A} \mathbf{u}} \left[a + \left(1 - \mathbf{u}\right) \mathbf{A}\right] - \mathbf{u} \left(1 - a\right)^2 \mathbf{D}\_{\alpha \mathbf{B}} \mathbf{e}^{\mathbf{B} \left(1 - \mathbf{u}\right)} \left(\frac{\partial \mathbf{u}}{\partial \mathbf{x}}\right)^2,\tag{23}$$

where

duration of the heating is dependent on the order of *m* mode, which can be clearly seen in the cases of the modes of higher order. For the modes of 5th and 6th order (Fig.18a), the increase of the heating duration in the first 25 h is accompanied by the increase of the birefringence. This effect can be explained by the generation of stresses in the deeper areas of the glass, which are reached by the admixture due to the diffusion widening of the doping area during the

On the basis of equation (22) and elastooptic constants (21) it can be calculated how the stresses at the glass surface change as a result of the heating processes. A graph of this relationship is shown in Fig.18 b. There can be seen that after a heating time of about 50 h, the value of stresses

**4. Repeatability of the technological effects of diffusion manufacturing** 

**5. Repeatability of the technological effects of diffusion manufacturing**

(a) (b) Fig.18 Refractive birefringence (at the glass surface) and modal birefringence depending on the time of the heating process Theat=(399 ± 1)°C (a). Changes in stresses at the glass surface in the heating

**Figure 18.** Refractive birefringence (at the glass surface) and modal birefringence depending on the time of the heating

use, the predictable and repeatable results are obtained. Those effects are the changes in glass refraction, which are of a gradient character. They are described by the refractive index profiles. Determination of the refractive index profiles is easy to implement in the case of planar waveguides. For this purpose a waveguide method with the use of selective excitation of modes using a prism coupler is used [33]. In this method, the measurements of synchronous angles are used, for which excitations of waveguide modes are obtained. Such measurements are made using a goniometer. Precision of the measurements of these angles is at the level of several tens of seconds. Based on the measured synchronous angles, the effective refractive indices of the waveguide modes are calculated. The uncertainties of their determination are at the level of 10-4. Based on the set of effective refractive indices of modes, a reconstruction of the refractive index profile of the waveguide is performed. For this purpose the procedure proposed in 1976 by White and Heidrich [45] is used. This procedure is based on the mode equation of the waveguide. Its

The ion exchange in glass method can be considered as a technological method. With its use, the predictable and repeatable results are obtained. Those effects are the changes in glass refraction, which are of a gradient character. They are described by the refractive index profiles. Determination of the refractive index profiles is easy to implement in the case of planar waveguides. For this purpose a waveguide method with the use of selective excitation of modes using a prism coupler is used [33]. In this method, the measurements of synchronous angles are used, for which excitations of waveguide modes are obtained. Such measurements are made using a goniometer. Precision of the measurements of these angles is at the level of several tens of seconds. Based on the measured synchronous angles, the effective refractive

The ion exchange in glass method can be considered as a technological method. With its

The control of technological processes of diffusion doping of glasses in the ion exchange

processes was proposed in the work [33] (p.181). The described method is used in producing one-

**processes of gradient changes in the refractive index of glasses** 

**processes of gradient changes in the refractive index of glasses**

process Theat=(399 ± 1)°C (a). Changes in stresses at the glass surface in the heating processes (b).

specific application is shown in [33] (p.169).

at the glass surface decreases to half its value after the diffusion process.

heating of the waveguide structure.

128 Ion Exchange - Studies and Applications

processes (b).

$$\alpha = 1 - \frac{\mathbf{D}\_{0\mathbf{A}}}{\mathbf{D}\_{0\mathbf{B}}} \cdot \exp\left[\mathbf{u}\left(\mathbf{A} + \mathbf{B}\right) - \mathbf{B}\right] \tag{24}$$

In the above equations, u(x) is a function describing the distribution of normalized concen‐ tration of admixture ions introduced into the glass. D0A,D0B, A, and B are coefficients that describe the functional dependence of diffusion coefficients of admixture ions DA(u) and the modifier DB(u) on normalized concentration of admixture introduced into the glass. These functions have the form [33]:

$$\mathbf{D}\_{\mathsf{A}}\left(\mathbf{u}\right) = \mathbf{D}\_{\mathsf{0A}}\mathbf{e}^{\mathsf{A}\mathsf{u}} \; \prime \quad \mathbf{D}\_{\mathsf{B}}\left(\mathbf{u}\right) = \mathbf{D}\_{\mathsf{0B}}\mathbf{e}^{\mathsf{B}\left(\mathbf{1} \cdot \mathbf{u}\right)} \tag{25}$$

The dependence of coefficients D0A and D0B on temperature is described by Arrhenius equa‐ tions:

$$\mathbf{D}\_{\rm oi} \left( \mathbf{T} \right) = \mathbf{D}\_{\rm oi}^{\*} \cdot \exp \left( -\frac{\Delta \mathbf{Q}\_{i}}{\mathbf{RT}} \right) \qquad \left( \mathbf{i} = \mathbf{A}, \mathbf{B} \right), \tag{26}$$

where ∆Q<sup>i</sup> - activation energy of the i-th ion type, R - universal gas constant.

The function u(x,t), which is a solution of the equation (23), transforms into one-dimensional refractive index profile n(x,t) according to the relation (5):

$$\mathbf{n}\left(\mathbf{x},\mathbf{t}\right) = \mathbf{n}\_{\flat} + \boldsymbol{\Delta}\mathbf{n}\_{\ast} \cdot \mathbf{u}\left(\mathbf{x},\mathbf{t}\right) \tag{27}$$

Thus, solving the diffusion equation is carried out in parallel with the realization of the diffusion process. The diffusion coefficients appearing in this equation are calculated at points in time tp based on the knowledge of their temperature dependencies. In this way the course of the process temperature Tp = T(tp) by the diffusion coefficients D0A(Tp) and D0B(Tp) affects the form of solution u(x,tp) of the diffusion equation (23). On the basis of equation (27) the dependence of function describing the refraction distribution in the glass n(x,tp) is obtained. In turn, with the use of equation (28), the effective refractive indices of the modes Nm(tp) corresponding with the refractive index profile n(x,tp) are calculated (for the assumed wave‐ length and state of polarization). The modal equation in the case of planar optical waveguides with a monotonic refractive index profile has the form:

$$\mathbf{k}\_{0} \int\_{0}^{\mathbf{x}\_{m}} \sqrt{\mathbf{n}^{2} \left(\mathbf{x}, \mathbf{t}\_{p}\right) - \mathbf{N}\_{m}^{2} \left(\mathbf{t}\_{p}\right)} \, d\mathbf{x} = \pi \left(\mathbf{m} + \frac{1}{4}\right) + \operatorname{arcctg}\left(\mathbf{r} \sqrt{\frac{\mathbf{N}\_{m}^{2} \left(\mathbf{t}\_{p}\right) - \mathbf{n}\_{c}^{2}}{\mathbf{n}\_{s}^{2} - \mathbf{N}\_{m}^{2} \left(\mathbf{t}\_{p}\right)}}\right) \,\tag{28}$$

where k0 - wave number of an electromagnetic wave in a free space, xm - position of the turning point of the m-th mode, n(x,tp) - refractive index profile of the waveguide, Nm(tp) - effective refractive index of the m-th mode, nc - refractive index of the ambient (coverage) of the waveguide, ns - refractive index of the waveguide at the glass surface, *r* = 1 (for TE polarization), *r* = (ns/nc) 2 (for TM polarization).

Visualization of the refractive index profile n(x,tp) and the resulting set of the effective refractive indices of modes Nm(tp) provide a direct control of the diffusion process. After the process, the duration of which was tdiff, a registered final refractive index profile of the glass n(x,tdiff) and a set of effective refractive indices {Nm(tdiff): *m* = 0,1...M-1} (M - number of modes) are obtained. Figure 19 schematically shows the principle of control of diffusion processes described earlier [46].

During the diffusion process control using the method mentioned earlier, the heat uniformity of the source of admixture has to be ensured. This is achieved by continuous mixing of the contents of the crucible. Thermocouple used to the measurements of the process temperature should be as close as possible to the glass substrate. The fulfillment of these conditions in the implemented technological processes was provided by a special handle made of silica glass (for details see [33] p.182).

In the continuation of this section the results being comparisons of effective refractive indices were presented. These comparisons relate to the effective refractive indices of modes calculated (Ncalc) during the diffusion process control with the results obtained from the measurements (Nmeas) of produced waveguides. The quantity of such comparisons are the absolute values of the differences of these values |Ncalc - Nmeas| calculated for each row of mode *m*. They were made for three types of glass substrates and two kinds of admixture ions. The glass substrates were: soda-lime (of Menzel-Glasser company), BK-7 (of Schott company), and Pyrex (Borosi‐ licate 33 of Corning company). The used admixture ions were silver ions Ag+ (source: silver

Producing the Gradient Changes in Glass Refraction by the Ion Exchange Method — Selected Aspects http://dx.doi.org/10.5772/60641 131

**Figure 19.** The principle of the diffusion process control in real time based on temperature measurements.

nitrate AgNO3) and potassium ions K+ (source: potassium nitrate KNO3). Table 5 summarizes the types of glass and the admixture ions, for which the temperature characteristics of diffusion coefficients were designated. The measurements of effective refractive indices of waveguide modes were performed for a wavelength *λ* = 677 nm (TE polarization).


**Table 5.** The types of glass and admixture ions.

n x,t = n + n u x,t ( ) b s D × ( ) (27)

( )

*t*

*p*

(source: silver

*p*

*t*

Thus, solving the diffusion equation is carried out in parallel with the realization of the diffusion process. The diffusion coefficients appearing in this equation are calculated at points in time tp based on the knowledge of their temperature dependencies. In this way the course of the process temperature Tp = T(tp) by the diffusion coefficients D0A(Tp) and D0B(Tp) affects the form of solution u(x,tp) of the diffusion equation (23). On the basis of equation (27) the dependence of function describing the refraction distribution in the glass n(x,tp) is obtained. In turn, with the use of equation (28), the effective refractive indices of the modes Nm(tp) corresponding with the refractive index profile n(x,tp) are calculated (for the assumed wave‐ length and state of polarization). The modal equation in the case of planar optical waveguides

( ) ( ) ( )

æ ö - æ ö ç ÷ - = ++ ç ÷ ç ÷ è ø - è ø ò (28)

where k0 - wave number of an electromagnetic wave in a free space, xm - position of the turning point of the m-th mode, n(x,tp) - refractive index profile of the waveguide, Nm(tp) - effective refractive index of the m-th mode, nc - refractive index of the ambient (coverage) of the waveguide, ns - refractive index of the waveguide at the glass surface, *r* = 1 (for TE polarization),

Visualization of the refractive index profile n(x,tp) and the resulting set of the effective refractive indices of modes Nm(tp) provide a direct control of the diffusion process. After the process, the duration of which was tdiff, a registered final refractive index profile of the glass n(x,tdiff) and a set of effective refractive indices {Nm(tdiff): *m* = 0,1...M-1} (M - number of modes) are obtained. Figure 19 schematically shows the principle of control of diffusion processes

During the diffusion process control using the method mentioned earlier, the heat uniformity of the source of admixture has to be ensured. This is achieved by continuous mixing of the contents of the crucible. Thermocouple used to the measurements of the process temperature should be as close as possible to the glass substrate. The fulfillment of these conditions in the implemented technological processes was provided by a special handle made of silica glass

In the continuation of this section the results being comparisons of effective refractive indices were presented. These comparisons relate to the effective refractive indices of modes calculated (Ncalc) during the diffusion process control with the results obtained from the measurements (Nmeas) of produced waveguides. The quantity of such comparisons are the absolute values of the differences of these values |Ncalc - Nmeas| calculated for each row of mode *m*. They were made for three types of glass substrates and two kinds of admixture ions. The glass substrates were: soda-lime (of Menzel-Glasser company), BK-7 (of Schott company), and Pyrex (Borosi‐

licate 33 of Corning company). The used admixture ions were silver ions Ag+

mx 2 2 m c 2 2

0 m 2 2 0 s m N n <sup>1</sup> k n x,t N dx m arctg r , <sup>4</sup> n N

p

with a monotonic refractive index profile has the form:

*p p*

(for TM polarization).

*r* = (ns/nc)

2

130 Ion Exchange - Studies and Applications

described earlier [46].

(for details see [33] p.182).

*t*

Table 6 summarizes a comparison of the waveguides produced in the soda-lime glass. In this glass the kinetics of Ag+ ↔ Na+ ion exchange is the highest among the other glass-admixture systems. A planar 3-modes waveguide is formed during a diffusion process with duration of 4 min. For the Ag+ ↔ Na+ ion exchange in this glass the largest absolute differences |Ncalc - Nmeas| here are of the order of 10-3.


**Table 6.** Diffusion processes in soda-lime glass. Ion exchange Ag+ ↔ Na+ . Source of admixture ions: AgNO3. *λ* = 677 nm, polarization TE.

The diffusion processes of K+ ↔ Na+ ion exchange are the slowest in the BK-7 glass. Table 7 shows a comparison of the effective refractive indices of the modes for diffusion processes lasting from 48 h to 216 h. The temperatures of these processes were approximately 400°C. In these cases, the differences of effective refractive indices of modes are of the order of 10-4. The

same differences occur in the control of the heating processes carried out in BK-7 glass. Table 8 presents the results referring to the heating processes for a waveguide produced in the preliminary diffusion process: tdiff = 74.3h, Tave = 401.2°C then subjected to heating. The temperatures of heating processes were at 445°C. The durations of the heating processes were 1 h, 2 h, and 4 h respectively.

Table 6 summarizes a comparison of the waveguides produced in the soda-lime glass. In this glass the kinetics of Ag+ ↔ Na+ ion exchange is the highest among the other glass-admixture systems. A planar 3-modes waveguide is formed during a diffusion process with duration of

**tdiff = 3h, Tave = 288°C tdiff = 4h, Tave = 280°C tdiff = 5h, Tave = 260.4°C Ncalc Nmeas |Ncalc Nmeas| Ncalc Nmeas |Ncalc Nmeas| Ncalc Nmeas |Ncalc Nmeas|**

 1.5924 1.5939 0.0015 1.5926 1.5938 0.0012 1.5906 1.5905 0.0001 1.5823 1.5833 0.0010 1.5826 1.5833 0.0007 1.5794 1.5786 0.0008 1.5744 1.5752 0.0008 1.5748 1.5752 0.0004 1.5702 1.5696 0.0006 1.5675 1.5682 0.0007 1.5679 1.5683 0.0004 1.5621 1.5614 0.0007 1.5611 1.5619 0.0008 1.5616 1.5620 0.0004 1.5544 1.5537 0.0007 1.5551 1.5558 0.0007 1.5555 1.5559 0.0004 1.5468 1.5463 0.0005 1.5492 1.5499 0.0007 1.5497 1.5501 0.0004 1.5392 1.5388 0.0004 1.5433 1.5440 0.0007 1.5438 1.5442 0.0004 1.5313 1.5313 0.0000 1.5373 1.5382 0.0009 1.5379 1.5384 0.0005 1.5235 1.5233 0.0002 1.5314 1.5323 0.0009 1.5319 1.5326 0.0007 1.5163 1.5162 0.0001

**Short time diffusion processes**

0 1.5805 1.5811 0.0006 1.5751 1.5760 0.0009 1.5712 1.5690 0.0022 1 1.5615 1.5621 0.0006 1.5513 1.5520 0.0007 1.5430 1.5389 0.0041 2 1.5454 1.5458 0.0004 1.5295 1.5310 0.0015 1.5154 1.5109 0.0045

The diffusion processes of K+ ↔ Na+ ion exchange are the slowest in the BK-7 glass. Table 7 shows a comparison of the effective refractive indices of the modes for diffusion processes lasting from 48 h to 216 h. The temperatures of these processes were approximately 400°C. In these cases, the differences of effective refractive indices of modes are of the order of 10-4. The

**tdiff = 15', Tave = 300°C tdiff = 8', Tave = 300°C tdiff = 4', Tave=302°C Ncalc Nmeas |Ncalc Nmeas| Ncalc Nmeas |Ncalc Nmeas| Ncalc Nmeas |Ncalc Nmeas|**

↔ Na+

. Source of admixture ions: AgNO3. *λ* = 677

10 1.5254 1.5265 0.0011 1.5259 1.5268 0.0009 11 1.5196 1.5206 0.0010 1.5201 1.5209 0.0008 12 1.5143 1.5151 0.0008 1.5147 1.5153 0.0006

ion exchange in this glass the largest absolute differences |Ncalc -

4 min. For the Ag+ ↔ Na+

132 Ion Exchange - Studies and Applications

**m**

**m**

3 1.5294 1.5297 0.0003 4 1.5140 1.5143 0.0003

nm, polarization TE.

**Table 6.** Diffusion processes in soda-lime glass. Ion exchange Ag+

Nmeas| here are of the order of 10-3.


**Table 7.** Diffusion processes in BK-7 glass. Ion exchange K+ ↔ Na+ . Source of admixture ions: KNO3. *λ* = 677 nm, polarization TE.

Table 9 shows the results referring to the Pyrex glass (Borosilicate 33). Also in this case the differences of calculated and measured effective refractive indices of modes are of the order of 10-4. The kinetics of Ag+ ↔ Na+ exchange processes in this glass is moderate. The singlemode waveguides are formed during 1 h and at a temperature of about 290°C.


**Table 8.** Heating processes in BK-7 glass. Ion exchange K+ ↔ Na+ . *λ* = 677 nm, polarization TE.

The results of comparisons of calculated and measured effective refractive indices confirm the effectiveness of the method of control of the diffusion processes. It is based on the best possible determined temperature dependence of the diffusion coefficients of exchanged ions for the given glass-admixture system. The described method allows to guarantee the repeatability of effective refractive indices at the level of ΔN ~10-3.

This method can also be used to control the production of two- and three-dimensional structures in gradient glasses.


**Table 9.** Diffusion processes in Pyrex glass. Ion exchange Ag+ ↔ Na<sup>+</sup> . Source of admixture ions: AgNO3. *λ* = 677 nm, polarization TE.
