**7. Destruction mechanism of inner cladding layer of HAF**

The internal core space of an HAF heated to an 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 an internal core temperature of about 10,000 K and an internal pressure of about 10,000 atmospheres (1 GPa) at the time of fiber fuse evolution [72].

In HAF2, *Dhole*/2 = 9.0 *µ*m and *rc* = 4.5 *µ*m. Therefore, the thickness *h* of the first cladding layer was 4.5 *µ*m before the core was heated to a high temperature. When the core is heated to a high temperature and the internal pressure *p* reaches 1 GPa, the first cladding layer is partially melted by heat transmitted from the core and *h* becomes smaller than its initial value (4.5 *µ*m).

We consider a tensile stress *σθ* acting on the inner wall of the first cladding layer with thickness *h*, as shown in Figure 24.

**Figure 24.** Tensile stress acting on inner wall of first cladding layer in HAF.

*σθ* is related to *p* by the following expression [96]:

26 Optical Fiber

SMF.

for HAF2 was ≃ 80 *µ*m.

about 120 *µ*m, and then stopped.

results of which are described below.

[51].

On the other hand, in HAF2, a large *P*<sup>0</sup> value is required to obtain sufficient accumulated heat for both the generation and propagation of a thermal wave, compared with that in the

Several researchers observed the dynamics of fiber fuse termination near a splice point

Takenaga *et al.* reported that when a laser light of *P*<sup>0</sup> = 8.1 W and *λ*<sup>0</sup> = 1.55 *µ*m was input into HAF2+, a fiber fuse was generated at the splice point between the HAF and the SMF, propagated about 100 *µ*m in the direction of the light source, and then stopped [42]. Kurokawa and Hanzawa [46] defined the length between the splice point and the termination

Similarly, Kurokawa and Hanzawa investigated the power dependence of both the propagation velocity *vf* for the SMF and *Lp* for HAF2 at *λ*<sup>0</sup> = 1.48 *µ*m [46]. When a laser light of *P*<sup>0</sup> = 8.1 W was input into HAF2, the observed *vf* for the SMF was 1.1 m/s and *Lp*

As shown in Figure 19, even if a high power of 9 W was input into HAF2, a fiber fuse with a high peak temperature of about 80,000 K was generated at the center of the heated core, but it did not propagate forward the light source. This phenomenon, in which the propagation of a fiber fuse is controlled, is in good agreement with the experimental results observed by

On the other hand, Kurokawa and Hanzawa reported that *vf* = 1.3 m/s and *Lp* = 110 *µ*m when a laser light of *P*<sup>0</sup> = 12.0 W (6.0 W at both 1.48 *µ*m and 1.55 *µ*m) was input into HAF2 [46]. This behavior for HAF2 cannot be explained by our calculation described above. As shown in Figure 22, a fiber fuse, generated at the splice point between the HAF2 and the SMF, was expected to maintain propagation in the direction of the light source when *Ith* = 12.2 MW/cm2 (*P*<sup>0</sup> = 12.0 W). However, it was reported that the fiber fuse propagated only

Furthermore, they reported that the hole part in the HAF disappeared in the domain in which the fiber fuse penetrated [46]. They considered that the plasma density of the core decreased in connection with the disappearance of the hole part and that the propagation of a fiber fuse can be controlled even if a high power of 10 W order is input into an HAF. In practice, they observed the dynamics of fiber fuse termination near the splice point between HAF2 and the SMF and found that the termination was accompanied by the evolution of a gas jet in the case of *P*<sup>0</sup> = 12 W [48] and 18.1 W (6.6 W at 1.48 *µ*m and 11.5 W at 1.55 *µ*m)

In order for the hole part to disappear with the gas jet, it is necessary for the (first) cladding

We considered the destruction of the cladding layer using our heat conduction model, the

The internal core space of an HAF heated to an elevated temperature is shielded from the external conditions. Thus, with increasing temperature, the internal pressure increases and

layer inscribed in this hole part to be destroyed by the incident high power.

**7. Destruction mechanism of inner cladding layer of HAF**

between a HAF and an SMF by using a high-speed camera [42], [46], [48], [51].

point of fiber fuse as a penetration length *Lp*. In this case, *Lp* ≃ 100 *µ*m.

Takenaga *et al.* and Kurokawa and Hanzawa [46] for HAF2.

$$\sigma\_{\theta} = \frac{D\_{\text{hole}}^2 - 2D\_{\text{hole}}h + 2h^2}{2D\_{\text{hole}}h - 2h^2} \cdot p. \tag{30}$$

*σθ* increases with increasing *p*. If *σθ* exceeds the ideal fracture strength *σ*<sup>0</sup> of the silica glass, a crack will generate on the inner wall of the first cladding layer.

On the other hand, it is well known for various solid materials that the *σ*<sup>0</sup> value is related to the Young's modulus *E* of the material by the following equation [97]:

$$
\sigma\_0 \approx E/10.\tag{31}
$$

By using Eq. (31) and *E* = 73 GPa for silica glass, we can estimate *σ*<sup>0</sup> to be approximately 7.3 GPa.

For the HAF2, we estimated the relationship between *h* and the tensile stress *σθ* acting on the inner wall of the first cladding layer for *p* = 1 GPa using Eq. (30). The calculation result is shown in Figure 25.

**Figure 25.** Thickness vs tensile stress acting on inner wall of cladding layer in HAF2.

*σθ* increases with decreasing *h*. It was found that when *h* decreases to a critical thickness *h*<sup>0</sup> (∼ 1.2 *µ*m), *σθ* becomes to *σ*<sup>0</sup> (∼ 7.3 GPa). That is, *σθ* becomes larger than *σ*<sup>0</sup> when *h* < *h*0. This means that the inner wall of the first cladding layer will crack and be destroyed when *h* < *h*0.

Silica glass has a melting temperature of *Tm* = 1,973 K. If solid silica glass is heated above *Tm*, it will become a liquid "melt" and its mechanical properties, such as tensile tolerance, will be lost.

If the heat conduction model discussed above is used, the temperature distribution along the internal radial direction can be estimated for the HAF2.

We can estimate the radial distance (equivalent to *h* in Figure 24) from the outer wall of the first cladding layer with temperature *Ta* to an inner point at which *T* reaches *Tm* using the heat conduction model. Then, the minimum power of incident laser light necessary for *h* < *h*<sup>0</sup> to be satisfied can be determined by estimating the *h* value when laser light with various powers enters the HAF2.

For incident laser powers *P*<sup>0</sup> of 4–10 W with *λ*<sup>0</sup> = 1.55 *µ*m, the temperature distribution in the first cladding layer as a function of the radial distance *h*′ was calculated for the HAF2. The result is shown in Figure 26.

The horizontal axis *h*′ in Figure 26 represents the distance between the outer wall of the first cladding layer and an inner point located closer to the core center.

When *P*<sup>0</sup> is 4 W or less, the temperature *T* in the cladding layer is lower than *Tm*, and the cladding layer is not destroyed.

On the other hand, when *<sup>P</sup>*<sup>0</sup> is 5 W and above, *<sup>T</sup>* increases with increasing *<sup>h</sup>*′ . When *P*<sup>0</sup> = 10 W, *T* reaches ∼ *Tm* at *h*′ ∼ 1.2 *µ*m. This value of *h*′ (∼ 1.2 *µ*m) is the same as the threshold value of *h*<sup>0</sup> (∼ 1.2 *µ*m) for crack generation on the inner wall of the cladding layer. Therefore, the destruction of the cladding layer is predicted when laser light with *P*<sup>0</sup> ≥ 10 W enters the HAF2.

**Figure 26.** Temperature field in first cladding layer of HAF2 when *P*<sup>0</sup> = 4–10 W at *λ*<sup>0</sup> = 1.55 *µ*m.

28 Optical Fiber

*h* < *h*0.

lost.

HAF2.

Elongation stress

0

**Figure 25.** Thickness vs tensile stress acting on inner wall of cladding layer in HAF2.

internal radial direction can be estimated for the HAF2.

cladding layer and an inner point located closer to the core center.

On the other hand, when *<sup>P</sup>*<sup>0</sup> is 5 W and above, *<sup>T</sup>* increases with increasing *<sup>h</sup>*′

various powers enters the HAF2.

The result is shown in Figure 26.

cladding layer is not destroyed.

01234 Thickness h (µm)

h0

*σθ* increases with decreasing *h*. It was found that when *h* decreases to a critical thickness *h*<sup>0</sup> (∼ 1.2 *µ*m), *σθ* becomes to *σ*<sup>0</sup> (∼ 7.3 GPa). That is, *σθ* becomes larger than *σ*<sup>0</sup> when *h* < *h*0. This means that the inner wall of the first cladding layer will crack and be destroyed when

Silica glass has a melting temperature of *Tm* = 1,973 K. If solid silica glass is heated above *Tm*, it will become a liquid "melt" and its mechanical properties, such as tensile tolerance, will be

If the heat conduction model discussed above is used, the temperature distribution along the

We can estimate the radial distance (equivalent to *h* in Figure 24) from the outer wall of the first cladding layer with temperature *Ta* to an inner point at which *T* reaches *Tm* using the heat conduction model. Then, the minimum power of incident laser light necessary for *h* < *h*<sup>0</sup> to be satisfied can be determined by estimating the *h* value when laser light with

For incident laser powers *P*<sup>0</sup> of 4–10 W with *λ*<sup>0</sup> = 1.55 *µ*m, the temperature distribution in the first cladding layer as a function of the radial distance *h*′ was calculated for the HAF2.

The horizontal axis *h*′ in Figure 26 represents the distance between the outer wall of the first

When *P*<sup>0</sup> is 4 W or less, the temperature *T* in the cladding layer is lower than *Tm*, and the

W, *T* reaches ∼ *Tm* at *h*′ ∼ 1.2 *µ*m. This value of *h*′ (∼ 1.2 *µ*m) is the same as the threshold value of *h*<sup>0</sup> (∼ 1.2 *µ*m) for crack generation on the inner wall of the cladding layer. Therefore, the destruction of the cladding layer is predicted when laser light with *P*<sup>0</sup> ≥ 10 W enters the

. When *P*<sup>0</sup> = 10

σ0

Inner pressure: 1GPa

5

10

15

20

25

σθ(GPa)

> If a thermal wave propagates along the fiber axis, because of the pressure difference of about 10,000 atmospheres between the internal core and the air hole layer, the destruction of the cladding layer will change the propagation direction of the thermal wave from the axial direction to the radial direction. As a result, the thermal wave will propagate toward the air hole layer with a low pressure (about 1 atmosphere). This is the reason for the observation of a gas jet reported by Kurokawa *et al.* [48].

> When the thermal wave propagates into the air hole layer, the hole layer disappears as a result of thermal heating and the core temperature decreases owing to the departure of the thermal wave. Such cooling of the core will prevent the generation of a new thermal wave in the core, and the propagation of the thermal wave will stop at this time.

> Kurokawa and Hanzawa [46] reported that when laser light with a high power entered HAF2, the increase in penetration length started at an incident power of 8 W.

> As described above, if the incident power increases from 9 W to 10 W, the cladding layer is destroyed and the direction of propagation of the thermal wave changes to the radial direction, and the air hole layer vanishes owing to the melting of the cladding layer.

> There is a slight time delay *τ* for an crack propagating from the inner surface to the outer surface of the first cladding layer. The crack generated on the inner surface of the first cladding layer grows and propagates to the outer surface of the cladding layer. When the crack reaches the outer surface, the first cladding layer is destroyed because of the pressure difference of about 10,000 atmospheres between the internal core and the air hole layer.

> The crack propagation rate *Vc* of the silica glass is related to the sonic rate *Vs* of the glass as follows [98]:

$$0 \le V\_{\mathcal{L}} \le 0.38V\_{\mathcal{s}}.\tag{32}$$

The value of *Vs* for silica glass is 5,570 m/s. For a crack propagating through a small length of ∼ 1.2 *µ*m (= *h*0), the minimum time delay *τ<sup>m</sup>* was estimated to be

$$
\pi\_{\rm ll} \equiv \frac{h\_0}{0.38 V\_{\rm s}} = 5.7 \times 10^{-10} \text{s}.\tag{33}
$$

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 Hanzawa [46].

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 HAF2.

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 period damage tracks observed in HAF2.

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)

$$F = \frac{n\_0 \pi \kappa \omega\_0^2}{\alpha P l (\partial n/\partial T)}^2,\tag{34}$$

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 value.

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

The cause of mechanism of the long-period damage track has yet to be sufficiently clarified. It largely depends upon future multilateral studies.
