**2.1. Effect of SiO***x* **formation on absorption**

4 Optical Fiber

*µ*m [19].

*α* values.

was observed.

changed from *T*<sup>0</sup> to *T*<sup>0</sup> + 50 K. Furthermore, Kashyap *et al.* reported that the best fit between the experimental and theoretical fiber fuse velocities was obtained when the *α* value of the Ge-doped silica core at high temperatures of above *<sup>T</sup>*<sup>0</sup> was fixed to be 4.0 ×10<sup>4</sup> <sup>m</sup>−<sup>1</sup> at 1.064

Hand and Russell found that this phenomenon was initiated by the generation of large numbers of Ge-related defects at high temperatures of above about 1,273 K, and the *α* values at *λ*<sup>0</sup> = 0.5 *µ*m obtained at temperatures of below 1873 K were modeled quite accurately using an Arrhenius equation [5], [6]. By contrast, they reported that the best fit between the experimental and theoretical fiber fuse velocities was obtained when the *α* value of the Ge-doped silica core at 2,293 K was assumed to be 5.6 ×104 m−<sup>1</sup> at 0.5 *µ*m [5]. This large *α* value, however, could not be estimated using their Arrhenius equation [6]. As the focal length *F* of thermal lense effect is inversely as the *α* value [63], large *α* value of 5.6 ×10<sup>4</sup> m−<sup>1</sup> is necessary to obtain small *F* value of 10 *µ*m order, which is comparable with observed

Furthermore, Hand and Russell reported that the electrical conductivity *σ* of the fiber core increased with the temperature and that the hot spot at the fiber fuse center was plasmalike [5]. Kashyap considered that the large *α* values may be attributable to an increase in the *σ*

It is well known that silica glass is a good insulator at room temperature, and the electrical conductivity in silica glass below 1,073 K is due to positively charged alkali ions moving under the influence of an applied field [64], [65]. The ionic conduction in the glass is not

We previously investigated the optical absorption mechanism causing the increase in the *σ* value and reported the relationship between *σ* and *α* in silica glass at high temperatures of above 1,273 K [66]–[69]. It was found that the increase in loss observed at 1.064 *µ*m can be well explained by the electronic conductivity induced by the thermal ionization of a

However, the calculated *α* values resulting from the electronic conductivity at 1.064 *µ*m were of 102 m−<sup>1</sup> order at 2,873 K, about two orders smaller than the *α* values (1.0–4.0 ×104 m<sup>−</sup>1) reported by Kashyap *et al.* [4], [19]. Therefore, we need another mechanism to explain the increase in loss at high temperatures of above 2,273 K to account for the large (104 m−<sup>1</sup> order)

To satisfy this requirement, we proposed a thermochemical SiO*x* production model in 2004 [68], [69]. Using this model, we theoretically estimated large *α* values of 104 m−<sup>1</sup> order as a result of SiO*x* absorption at high temperatures of 2,800 K or above. This model was able to quantitatively explain the relation between the fiber fuse propagation velocity and the

On the other hand, since the parameters (particularly the light-absorbing parameter) used for the numerical simulation were not optimized, the shortcoming that the maximum temperature of the core center obtained by calculation became unusually high (10<sup>5</sup> K order)

We have improved this model by optimizing several parameters required for the numerical

incident laser-power intensity previously reported by other research institutions.

computation, and we proposed an improved model in 2014 [70].

′

centers.

Ge-doped silica core, and it is not directly related to the absorption of Ge *E*

interval and/or large front size of the cavities (see Appendix).

value of the fiber core at high temperatures of above *T*<sup>0</sup> [4].

related solely to optical absorption.

It has been reported that, at elevated temperatures, silica glass is thermally decomposed by the reaction [71]

$$\rm{SiO\_2} \rightleftharpoons \rm{SiO\_3} + \rm{(x/2)O\_2}.\tag{1}$$

Among the reductants of the silica generated by this pyrolysis reaction with the formula SiO*x*, the most thermally stable material is SiO (*x* = 1).

The internal core space heated at the elevated temperature is shielded from the external conditions. Thus, with increasing temperature, the internal pressure increases and the internal volume decreases. Dianov *et al.* reported that the internal core temperature is about 10,000 K and the internal pressure is about 10,000 atmospheres at the time of fiber fuse evolution [72]. It is thought that SiO*x* generated under the high-temperature and high-pressure conditions is densely packed into the internal core space because it is not allowed to expand, and it exists in a liquidlike form. For this reason, it is thought that the optical absorption spectrum of the high-density SiO*x* in the core space is similar to that of solid SiO*x*.

Philipp reported that the optical absorption spectrum of a SiO*x* film is similar to that of amorphous Si because of the many Si–Si bonds in a SiO*x* film, and that the absorption coefficient *<sup>α</sup>*SiO for SiO (*<sup>x</sup>* = 1) near the threshold energy should be about one-twentieth of that for amorphous Si *α*Si [73]. Furthermore, Philipp estimated the theoretical concentration *fSi* of Si–(Si4) when the five possible tetrahedral conformations centering on Si, Si–(Si4), Si–(Si3O), Si–(Si2O2), Si–(SiO3), and Si–(O4), were completely distributed at random within amorphous SiO*<sup>x</sup>* [74]. The relationship between *fSi* and *x* for SiO*<sup>x</sup>* is illustrated in Figure 4. According to Figure 4, *fSi* for a SiO (*x* = 1) film is 6.25% (= 1/16). This is very close to the factor by which the *α* value was reduced (about 1/20) as reported by Philipp [73].

The optical absorption near the absorption edge of a SiO*x* film is dominated by that of amorphous Si with the Si–(Si4) tetrahedral conformation, and the optical absorption coefficient *α*SiO*<sup>x</sup>* near the absorption edge of a SiO*<sup>x</sup>* film can be calculated by multiplying the value of *αSi* for amorphous Si by *fSi*. That is, if the production rate of SiO*<sup>x</sup>* given by Eq.

**Figure 4.** Relative content of Si–(Si4) in SiO*x*.

(1) is denoted by *g*SiO*<sup>x</sup>* , then *α*SiO*<sup>x</sup>* at temperature *T* is given by

$$
\mathfrak{a}\_{\text{SiO}\_x}(T) = \mathfrak{g}\_{\text{SiO}\_x}(T) f\_{\text{Si}}(\mathfrak{x}) \mathfrak{a}\_{\text{Si}}(T). \tag{2}
$$
