**5. Heat conduction modeling of HAF**

18 Optical Fiber

Tav (x103K)

in the core center of the HAF.

dh

**Figure 13.** Heat conduction model of HAF [44].

Dhole

2rc

**Figure 12.** Average temperature of radiation zone vs. time after fiber fuse generation.

t (ms)

will occur between the heated solid inner surfaces of the holes and the gaseous fluid (air) with the low temperature of *T* = *Ta*. This is expected to affect the heat conduction behavior

Thus, we used the model proposed by Takara *et al.* [44] for heat conduction analysis, which includes heat transfer or radiative transfer between the inner surfaces of the holes and the

dh

First cladding

Dhole

Second cladding

2rc

Core

Hole

gaseous air in the HAF. The proposed model for the HAF is shown in Figure 13.

Core

Air hole

Cladding

Although bridges of silica glass exist between the holes in an actual HAF, to simplify the calculation, the first cladding layer was assumed to be a cylindrical layer of *Dhole* diameter, and the hole layer was treated as a gap of *dh* width, which was inserted between the first and second cladding layers (see Figure 13). The hole layer was assumed to be fulfilled with a silica-air mixture. In the hole layer, the volume ratio of a silica glass to an air was (1 - *γ*) : *γ*, where *γ*<sup>1</sup> is the occupancy of the 6-air holes in the cross section of HAF. Furthermore,

0 5 10 15 20 25 30

P = 2W

5760 K

We assume the HAF to have a radius of *rf* and to be in an atmosphere with temperature *T* = *Ta*. We also assume that part of the core layer of ∆*L* length is heated to a temperature of *T*0 *<sup>c</sup>* (> *Ta*) (see Figure 14).

**Figure 14.** Hot zone in core layer of HAF.

The optical absorption coefficient *α* of the core layer in a HAF is a function of temperature (*T*), and it increases with increasing *T* [70]. In the heating zone (called the "hot zone") shown in Figure 14, *α* is larger than in other parts of the core because of its high temperature *T*0 *<sup>c</sup>* (> *Ta*). Thus, as light propagates along the positive direction (away from the light source) in this zone, considerable amount of heat is produced by light absorption.

In the case of a heat source in part of the core layer, the nonsteady heat conduction equation for the temperature field *T*(*r*, *z*, *t*) in the HAF is given by Eq. (12). We can solve Eq. (12) using the explicit finite-difference method (FDM) under the boundary and initial conditions, described in section 3.3, and the additional boundary conditions as follows:

(1) At the outer surface (*r* = *Dhole*/2) of the first cladding layer, the amount of heat conducted per unit area (heat flux) is dissipated by radiative transfer or heat transfer to the air (*T* = *Ta*) in the hole layer as follows:

$$\begin{aligned} -\lambda \frac{\partial T}{\partial r}\bigg|\_{r=D\_{\text{hole}}/2} &= \gamma\_1 \sigma\_S \varepsilon\_\varepsilon \left(T^4 - T\_a^4\right) \\ &+ \gamma\_1 \frac{\lambda}{\delta r\_t} \left(T - T\_a\right), \end{aligned} \tag{28}$$

where *γ*<sup>1</sup> is the ratio of the total surface area of the 6-air holes existing in an actual HAF to the outer surface area of the first cladding layer, *δrt* is the thickness of the thermal boundary layer, and *δrt* = *x δr* is assumed in the HAF calculation, where *x* = 0.5.

(2) At the inner surface (*r* =*Dhole*/2 + *dh*) of the second cladding layer, the amount of heat conducted per unit area (heat flux) is dissipated by radiative transfer or heat transfer to the air (*T* = *Ta*) in the hole layer as follows:

$$\begin{aligned} -\lambda \left. \frac{\partial T}{\partial r} \right|\_{r=D\_{\text{hole}}/2+d\_{\text{h}}} &= \gamma\_2 \sigma\_S \varepsilon\_e \left( T^4 - T\_d^4 \right) \\ &+ \gamma\_2 \frac{\lambda}{\delta r\_t} \left( T - T\_d \right) . \end{aligned} \tag{29}$$

where *γ*<sup>2</sup> is the ratio of the total surface area of the 6-air holes existing in an actual HAF to the inner surface area of the second cladding layer.

The fiber parameters of the HAFs used for calculation were *dh* = *rc* = 4.5 *µ*m, which were used in the experiments reported by Hanzawa *et al.* [45]. *Rh* was set to 2, 3, and 4. In the following, HAFs with *Rh* = 2, 3, and 4 are called HAF2, HAF3, and HAF4, respectively.

The parameters *γ*, *γ*1, *γ*<sup>2</sup> for HAF2, HAF3, HAF4 are shown in Table 1.


**Table 1.** Parameters for HAFs.

The area in the numerical calculation has a length of 2*L* (= 4 cm) in the axial (*z*) direction and a width of 2*rf* (= 125 *µ*m) in the radial (*r*) direction. There were 28 and 2,000 divisions in the *r* and *z* directions, respectively, and we set the calculation time interval to 1 *µ*s. We assume that the hot zone is located at the center of the fiber (length 2*L*) and that the length ∆*L* of the hot zone is 40 *µ*m.

### **6. Simulation of fiber fuse in HAF**

20 Optical Fiber

air (*T* = *Ta*) in the hole layer as follows:

air (*T* = *Ta*) in the hole layer as follows:

<sup>−</sup> *<sup>λ</sup> <sup>∂</sup><sup>T</sup> ∂r r*=*Dhole*/2

layer, and *δrt* = *x δr* is assumed in the HAF calculation, where *x* = 0.5.

*r*=*Dhole*/2+*dh*

The parameters *γ*, *γ*1, *γ*<sup>2</sup> for HAF2, HAF3, HAF4 are shown in Table 1.

<sup>−</sup> *<sup>λ</sup> <sup>∂</sup><sup>T</sup> ∂r* 

the inner surface area of the second cladding layer.

**Table 1.** Parameters for HAFs.

hot zone is 40 *µ*m.

(1) At the outer surface (*r* = *Dhole*/2) of the first cladding layer, the amount of heat conducted per unit area (heat flux) is dissipated by radiative transfer or heat transfer to the

= *γ*1*σSǫ<sup>e</sup>*

+ *γ*<sup>1</sup> *λ δrt*

where *γ*<sup>1</sup> is the ratio of the total surface area of the 6-air holes existing in an actual HAF to the outer surface area of the first cladding layer, *δrt* is the thickness of the thermal boundary

(2) At the inner surface (*r* =*Dhole*/2 + *dh*) of the second cladding layer, the amount of heat conducted per unit area (heat flux) is dissipated by radiative transfer or heat transfer to the

= *γ*2*σSǫ<sup>e</sup>*

+ *γ*<sup>2</sup> *λ δrt*

where *γ*<sup>2</sup> is the ratio of the total surface area of the 6-air holes existing in an actual HAF to

The fiber parameters of the HAFs used for calculation were *dh* = *rc* = 4.5 *µ*m, which were used in the experiments reported by Hanzawa *et al.* [45]. *Rh* was set to 2, 3, and 4. In the following, HAFs with *Rh* = 2, 3, and 4 are called HAF2, HAF3, and HAF4, respectively.

> Type *γ γ*<sup>1</sup> *γ*<sup>2</sup> HAF2 0.300 1.50 1.00 HAF3 0.214 1.00 0.75 HAF4 0.167 0.75 0.60

The area in the numerical calculation has a length of 2*L* (= 4 cm) in the axial (*z*) direction and a width of 2*rf* (= 125 *µ*m) in the radial (*r*) direction. There were 28 and 2,000 divisions in the *r* and *z* directions, respectively, and we set the calculation time interval to 1 *µ*s. We assume that the hot zone is located at the center of the fiber (length 2*L*) and that the length ∆*L* of the

 *T*<sup>4</sup> − *T*<sup>4</sup> *a* 

 *T*<sup>4</sup> − *T*<sup>4</sup> *a* 

(*T* − *Ta*), (28)

(*T* − *Ta*), (29)

On the basis of the model described above, we calculated the generation and propagation behavior of a fiber fuse in HAFs with *dh* = *rc* = 4.5 *µ*m using the FDM.

For reference, we also carried out the calculation for an SMF which has *rc* = 4.5 *µ*m, a refractive index difference ∆ = 0.3%, and *Aeff* = 98.1318 × 10−<sup>12</sup> m2.

The calculation using the FDM was conducted in accordance with the procedure in [70]. In the calculation, we set the time interval *δt* to 1 *µ*s, the step size along the *r* axis *δr* to *rf* /14, the step size along the *z* axis *δz* to 20 *µ*m, respectively, and assumed that *T*<sup>0</sup> *c* = 2,923 K and *Ta* = 298 K.

We estimated the temperature field *T*(*r*, *z*) of HAFs and/or SMF at *t* = 10 ms after the incidence of laser light with wavelength *λ*<sup>0</sup> = 1.55 *µ*m and an initial power *P*<sup>0</sup> of 4 W. The calculation results for HAF2, HAF3, and HAF4 are shown in Figures 15–17, respectively, and the result for the SMF is shown in Figure 18.

**Figure 15.** Temperature field in HAF2 after 10 ms after when *P*<sup>0</sup> = 4 W at *λ*<sup>0</sup> = 1.55 *µ*m.

**Figure 16.** Temperature field in HAF3 after 10 ms when *P*<sup>0</sup> = 4 W at *λ*<sup>0</sup> = 1.55 *µ*m.

**Figure 17.** Temperature field in HAF4 after 10 ms when *P*<sup>0</sup> = 4 W at *λ*<sup>0</sup> = 1.55 *µ*m.

**Figure 18.** Temperature field in SMF after 10 ms when *P*<sup>0</sup> = 4 W at *λ*<sup>0</sup> = 1.55 *µ*m.

As shown in Figures 15–18, the core center temperature at the thermal peak position of HAF2, HAF3, HAF4, and SMF are 2,923 K, 8,756 K, 11,603 K, and 17,528 K, respectively. The papid rise in the temperature shown in HAF3, HAF4, and SMF initiates the fiber fuse phenomenon. In SMF, the propagation velocity *vf* of the fiber fuse was estimated to about 0.43 m/s from the peak shift distance (4.3 mm) per 10ms of the core center temperature shown in Figure 18. In HAF3 and HAF4, the propagation velocities of the fiber fuse were estimated to about 0.26 and 0.36 m/s, respectively, by using the same procedure as above.

On the other hand, in HAF2, there is no temperature increase in the core layer, as shown in Figure 15, and it means that a fiber fuse was not generated when *P*<sup>0</sup> = 4 W.

We investigated the temperature field *T*(*r*, 0) at the end (*z* = 0 mm) of the hot zone in HAF2 after 10 ms when *P*<sup>0</sup> = 4–10 W at *λ*<sup>0</sup> = 1.55 *µ*m. The calculated temperature fields are shown in Figure 19.

When the power of the light entering HAF2 increases from 4 W to 5 W, a thermal wave with a peak temperature of higher than 60,000 K was generated at the end of the hot zone, as

10.5772/58959

22 Optical Fiber

HAF4

SMF



0.5 1

r / r*<sup>f</sup>*

r / r*<sup>f</sup>*



As shown in Figures 15–18, the core center temperature at the thermal peak position of HAF2, HAF3, HAF4, and SMF are 2,923 K, 8,756 K, 11,603 K, and 17,528 K, respectively. The papid rise in the temperature shown in HAF3, HAF4, and SMF initiates the fiber fuse phenomenon. In SMF, the propagation velocity *vf* of the fiber fuse was estimated to about 0.43 m/s from the peak shift distance (4.3 mm) per 10ms of the core center temperature shown in Figure 18. In HAF3 and HAF4, the propagation velocities of the fiber fuse were estimated to about 0.26

On the other hand, in HAF2, there is no temperature increase in the core layer, as shown in

We investigated the temperature field *T*(*r*, 0) at the end (*z* = 0 mm) of the hot zone in HAF2 after 10 ms when *P*<sup>0</sup> = 4–10 W at *λ*<sup>0</sup> = 1.55 *µ*m. The calculated temperature fields are shown

When the power of the light entering HAF2 increases from 4 W to 5 W, a thermal wave with a peak temperature of higher than 60,000 K was generated at the end of the hot zone, as

**Figure 17.** Temperature field in HAF4 after 10 ms when *P*<sup>0</sup> = 4 W at *λ*<sup>0</sup> = 1.55 *µ*m.

**Figure 18.** Temperature field in SMF after 10 ms when *P*<sup>0</sup> = 4 W at *λ*<sup>0</sup> = 1.55 *µ*m.

and 0.36 m/s, respectively, by using the same procedure as above.

Figure 15, and it means that a fiber fuse was not generated when *P*<sup>0</sup> = 4 W.

T (×10<sup>3</sup> K)

in Figure 19.

T (×10<sup>3</sup> K)

**Figure 19.** Temperature field at the end of the hot zone in HAF2 after 10 ms when *P*<sup>0</sup> = 4–10 W at *λ*<sup>0</sup> = 1.55 *µ*m.

shown in Figure 19. Figure 20 shows the peak temperature change at the core center (*r* = 0 *µ*m) with passage of time after the incidence of laser light with *P*<sup>0</sup> = 5 W at *λ*<sup>0</sup> = 1.55 *µ*m.

**Figure 20.** Peak temperature at the core center vs. time after the incidence of laser light with *P*<sup>0</sup> = 5 W at *λ*<sup>0</sup> = 1.55 *µ*m.

The core center temperature at the end of the hot zone (*z* = 0 mm) changes abruptly to a large value of ≥ 50,000 K after 28 *µ*s.

Although a thermal wave with a peak temperature of higher than 60,000 K was generated at the end of the hot zone, it was found that this thermal wave did not increase in size or propagate in the negative *z* direction when *P*<sup>0</sup> increased from 5 W to 10 W, as shown in Figure 19. Such a "stationary" thermal wave was reported by Kashyap [14].

If *P*<sup>0</sup> is further increased to 11 W and above, the thermal wave increases in size and propagates in the negative *z* direction toward the light source, as shown in Figure 21.

**Figure 21.** Temperature field in HAF2 after 10 ms after when *P*<sup>0</sup> = 11 W at *λ*<sup>0</sup> = 1.55 *µ*m.

When *P*<sup>0</sup> = 11 W, the propagation velocities of the fiber fuse were estimated to about 0.38 m/s by using the same procedure as above.

We calculated the power intensity dependence of the propagation velocity *vp* for each HAF and the SMF at *λ*<sup>0</sup> = 1.55 *µ*m. The calculation results for the SMF, HAF2, HAF3, and HAF4 are shown in Figure 22.

**Figure 22.** Power intensity vs fiber-fuse propagation velocity for SMF, HAF2, HAF3, and HAF4.

It is clear that the propagation velocity of the HAF4 and/or HAF3 approaches that of the SMF with increasing power intensity.

The threshold power intensity *Ith* for HAF2, HAF3, HAF4, and SMF are shown in Table 2.


**Table 2.** Threshold power intensity for HAFs and SMF.

24 Optical Fiber

propagate in the negative *z* direction when *P*<sup>0</sup> increased from 5 W to 10 W, as shown in

If *P*<sup>0</sup> is further increased to 11 W and above, the thermal wave increases in size and propagates in the negative *z* direction toward the light source, as shown in Figure 21.

HAF2 P0 = 11 W


HAF2

0.5 1

r / r*<sup>f</sup>*

Figure 19. Such a "stationary" thermal wave was reported by Kashyap [14].


λ = 1.55 µm

When *P*<sup>0</sup> = 11 W, the propagation velocities of the fiber fuse were estimated to about 0.38

We calculated the power intensity dependence of the propagation velocity *vp* for each HAF and the SMF at *λ*<sup>0</sup> = 1.55 *µ*m. The calculation results for the SMF, HAF2, HAF3, and HAF4

0 2 4 6 8 10 12 14 16 18 20

HAF3

Power intensity (MW/cm2)

It is clear that the propagation velocity of the HAF4 and/or HAF3 approaches that of the

The threshold power intensity *Ith* for HAF2, HAF3, HAF4, and SMF are shown in Table 2.

**Figure 21.** Temperature field in HAF2 after 10 ms after when *P*<sup>0</sup> = 11 W at *λ*<sup>0</sup> = 1.55 *µ*m.

m/s by using the same procedure as above.

0.0

0.4

0.8

Velocity (m/s)

SMF with increasing power intensity.

SMF

HAF4

**Figure 22.** Power intensity vs fiber-fuse propagation velocity for SMF, HAF2, HAF3, and HAF4.

1.2

1.6

2.0

are shown in Figure 22.

T (×10<sup>3</sup> K)

> On the other hand, in HAF2, fiber fuse propagation begins at a threshold power intensity *Ith* of about 10.4 MW/cm<sup>2</sup> (*P*<sup>0</sup> = 10.2 W), which is much higher than that in HAF3 (about 3.3 MW/cm2) or HAF4 (about 2.7 MW/cm2), as shown in Table 2. The *Ith* of HAF2 is six times that of SMF.

> When laser light enters the hot zone of the core layer, heat is produced in the zone by optical absorption of the incident light. In HAF2, the heat generated by optical absorption is effectively dissipated by the heat transfer and radiative transfer between the inner surfaces of the hole layer and the gaseous air. The dissipation of heat in HAF2 is more effective than that in HAF3 or HAF4 because the first cladding layer in HAF2 is thinner than that in HAF3 or HAF4.

> Figure 23 shows a schematic view of the relationship between *P*<sup>0</sup> and the accumulated heat in the hot zone of the core layer for HAF2 and SMF.

**Figure 23.** Schematic view of incident laser power vs accumulated heat in hot zone for HAF2 and SMF.

When the light power is small, the heat generated in the hot zone flows into the adjacent cladding layer, and the rise in the temperature of the zone is prevented. However, if light with a threshold power *Pth* and above enters the hot zone, it becomes difficult for all the generated heat to escape into the cladding layer, and part of the heat accumulates in the hot zone.

In the SMF, the accumulated heat raises the temperature in the hot zone and a thermal wave, *i.e.*, fiber fuse, is generated at the center of the zone. The thermal wave then rapidly increases in size and starts to propagate toward the light source as shown in Figure 23.

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 SMF.

Several researchers observed the dynamics of fiber fuse termination near a splice point between a HAF and an SMF by using a high-speed camera [42], [46], [48], [51].

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

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* for HAF2 was ≃ 80 *µ*m.

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

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 about 120 *µ*m, and then stopped.

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) [51].

In order for the hole part to disappear with the gas jet, it is necessary for the (first) cladding layer inscribed in this hole part to be destroyed by the incident high power.

We considered the destruction of the cladding layer using our heat conduction model, the results of which are described below.
