components. In alkali borosilicate glasses, a model proposed by Dell et al. $ ^{[15]} $ has been widely used for the prediction of B4 fraction. In practice, however, it is difficult to predict B4 fraction due to the compositional change in a component other than $ R_{2}O $ , $ B_{2}O_{3} $ and $ SiO_{2} $ . As mentioned, glass structure is dependent on basicity, $ ^{[2]} $ and it is therefore expected that B4 fraction is expressed as a function of basicity. Then, the relation between B4 fraction and basicity was examined in alkali and alkaline-earth borosilicate glasses. $ ^{[16,17]} $

Fig. 5. Fraction of four-fold coordinated boron atoms given by (a) $ N_{4}=B4/B $ , (b) B4/(B+Si) and (c) B4/(B+Si×f) in $ Na_{2}O-CaO-B_{2}O_{3}-SiO_{2} $ system. $ ^{16)} $

Figure 5(a) shows the relations between conventional notation of B4 fraction, $ N_{4} $ and basicity, $ ^{16)} $ where $ N_{4} $ is given by the proportion of B4 to total boron content, B4/B. At constant $ SiO_{2}/B_{2}O_{3} $ ratios, maxima in $ N_{4} $ are seen commonly at around $ \Lambda(\chi)=0.55 $ , and $ N_{4} $ values at maxima are different between $ SiO_{2}/B_{2}O_{3} $ ratios. $ N_{4} $ value is dependent on $ SiO_{2}/B_{2}O_{3} $ ratio, and hence the proportion of B4 to total content of network forming atoms, B4/(B+Si) is examined. $ ^{16)} $ As shown in Fig. 5(b), the higher the $ SiO_{2}/B_{2}O_{3} $ ratio, the smaller the B4/(B+Si) value, which is opposite to Fig. 5(a). From Figs. 6(a) and 6(b), $ SiO_{2} $ seems to contribute surely but partly the formation of B4. Then, introducing a coefficient f, B4 fraction is expressed by B4/(B+Si×f). $ ^{16)} $ As shown in Fig. 5(c), B4/(B+Si×f) value is seemingly independent to $ SiO_{2}/B_{2}O_{3} $ ratio and is expressed by a polynomial function of $ \Lambda(\chi) $ . The coefficients f for various alkali and alkaline-earth oxides are given in Fig. 6. Except for $ Li_{2}O $ , a linear correlation between f and field strength, Z/r is confirmed, where Z and r are the charge and radius of a cation, respectively. The f values given in Fig. 6 are applicable to the mixed alkali and alkaline-earth glasses, $ ^{17)} $ indicating that the prediction of B4 fraction is possible even in the multi component glasses.

Fig. 6. Field strength, $ Z/r^{2} $ and the optimal coefficient f in B4/(B+Si×f) for the ternary borosilicate glasses. $ ^{[16]} $ Z and r indicate ionic charge and radius, respectively. Open circle: f/2 for alkaline-earth oxides.

### 4. Material and process design based on basicity

### 4.1 Compositional design in optical function

Optical absorption and emission of $ Er^{3+} $ ions change significantly according to glass composition. As shown in Fig. 7, optical absorption, $ \sigma_{a} $ at around 1.5 $ \mu $ m consists of broad magnetic dipole (MD) and sharp electric dipole (ED) transitions. $ ^{[18]} $ In particular, relative intensity of these transitions changes considerably in borate glass. In the usage as amplifier, glasses with broad absorption and emission peaks are desirable. When $ O^{2-} $ ions neighboring to $ Er^{3+} $ ions have less electron donating ability, that is, small basicity, probability of ED transition increases, resulting in wider line width. $ ^{[19]} $ Then, relation between the line width and basicity was examined. $ ^{[18]} $