**8. Conclusions**

30 Optical Fiber

Hanzawa [46].

period damage tracks observed in HAF2.

observed period (460 *µ*m) of the cavities [46], [48]–[50].

It largely depends upon future multilateral studies.

HAF2.

value.

*<sup>τ</sup><sup>m</sup>* <sup>≡</sup> *<sup>h</sup>*<sup>0</sup> 0.38*Vs*

This time delay is not dependent on the incident power, as indicated by Eq. (32), and the propagation velocity *vf* of the thermal wave increases with increasing incident power.

We assumed that *τ* for the HAF2 corresponded to the time (50–90 *µ*s) for rapid rise in the core temperature after the incidence of fiber fuse, which propagated in the SMF. If we assume that *τ* ∼ 80 *µ*s and the penetration length *Lp* is the product of *vf* and *τ*, the *Lp* values at a incident light of *P*<sup>0</sup> = 8.1 W and 12.0 W are 88 *µ*m and 104 *µ*m. These values are fairly good agreement with the experimental results (∼ 80 *µ*m and 110 *µ*m) reported by Kurokawa and

In closing, we should comment on the fiber fuse propagation with a long-period damage track, which was observed in the HAF2 with *dh* of smaller than *rc* [46], [48]–[50]. The most striking characteristic of this phenomenon is the long period (several 100 *µ*m order) of the cavities, which were generated by entering of a relatively low power (about 1.5–4.5 W) into

The long period damage tracks such as those in HAF2 were reported by Bufetov *et al.* [99]. They found that such a phenomenon was observed as a result of the interference of LP01 and LP02 modes excited in an optical fiber with a W-index profile. However HAF2 is a step-index optical fiber, and only LP01 (or TE11) mode can be excited in HAF2 at 1.48 and/or 1.55 *µ*m. Therefore, it is very difficult to consider that the mode interference is responsible for the long

Instead of the mode interference, we consider that this phenomenon may relate to a thermal lense effect [63], [101]. The focal length *F* of thermal lense effect is given by (see Appendix)

2

*<sup>F</sup>* <sup>=</sup> *<sup>n</sup>*0*πκω*<sup>0</sup>

*αPl*(*∂n*/*∂T*)

where *n*<sup>0</sup> (= 1.46) is the characteristic refractive index of the core layer, *P* is the incident power of the light reflecting from the cavity wall, which is estimated by Eq. (24). *ω*<sup>0</sup> (∼ 4.5 *µ*m) is the spot size radius of the laser beam when optical power in the optical fiber was assumed to take on Gaussian distribution, *∂n*/*∂T* (= 1.23 × 10−<sup>5</sup> K−<sup>1</sup> [5]) is the thermal coefficient of refractive index for silica glass, and *l* is the length of the heating core, where *α* exhibits large

As *F* is inversely as the product of *α* and *P*, large *F* value may be obtained by small *α* and/or *P* value, which is comparable with observed long period of the cavities. If we assume *λ* = 9.2 W/mK [5], *<sup>l</sup>* = 20 *<sup>µ</sup>*m, and *<sup>α</sup>* = 5 × 104 <sup>m</sup><sup>−</sup>1, the *<sup>F</sup>* value at *<sup>P</sup>* = 0.158 W (*P*<sup>0</sup> = 4.5 W) can be estimated to about 440 *µ*m by using Eq. (34). This value is fairly good agreement with the

The cause of mechanism of the long-period damage track has yet to be sufficiently clarified.

= 5.7 × 10<sup>−</sup>10s. (33)

, (34)

We investigated the unsteady thermal conduction status in a single-mode optical fiber by numerical computation in order to visualize the mode of fiber fuse propagation. We assumed that the vitreous silica optical fiber underwent pyrolysis at elevated temperatures to form SiO*<sup>x</sup>* (*x* ∼ 1). We also proposed a model in which the optical absorption coefficient of the core layer increased with increasing molar concentration of SiO*x*. By using the model, we calculated the temperature distribution in the fiber with the explicit finite-difference method. It was found that when a short core with 40 *µ*m length was heated to 2,923 K and a 2 W laser light (wavelength of 1.064 *µ*m) entered the core layer of an SMF-28 optical fiber, a thermal wave, *i.e.*, a fiber fuse, with a peak temperature of about 34,000 K was generated at the boundary of the heating region near the light source. The fiber fuse was enlarged and propagated toward the light source at a rate of about 0.54 m/s. The calculated propagation velocity of the fiber fuse was in agreement with the experimental value. Moreover, the average temperature of the radiated region of the core layer was less than 7,000 K at a time of 4 ms after the generation of the fiber fuse and gradually approached a temperature of about 5,700 K. The final average temperature was close to the experimentally reported values.

We evaluated the threshold power of fiber fuse propagation in hole-assisted fibers (HAFs) using the finite-difference method and the model proposed by Takara *et al.* The HAFs with ratios of hole-space distance to core diameter of 3 and 4 exhibited fiber fuse propagation when a 1.55 *µ*m laser with a power of 4 W was input into the core layer, as observed for an SM optical fiber.

On the other hand, the HAF with a ratio of hole-space distance to core diameter of 2 exhibited no fiber fuse generation or propagation when a 1.55 *µ*m laser with a power of 10 W was input into the core layer.

Furthermore, when the incident power was 5 W and above, the temperature of the central core increased owing to the absorption of a large amount of power, causing the melting of the first cladding layer adjacent to the heated core. Thus, the thickness of the first cladding layer decreased below the value at which the solid cladding layer could be maintained, and the cladding layer was destroyed following the disappearance of the air hole layer. The destruction of the cladding layer caused the direction of the thermal wave to change from the axial direction to the radial direction, stopping the propagation of the thermal wave. By using this phenomenon, Kurokawa and Hanzawa proposed a novel fiber fuse terminotor composed of a short length of HAF [100].
