**2.2. Production rate of SiO***<sup>x</sup>*

The most thermally stable compound in SiO*x* generated via Eq. (1) is SiO (*x* = 1). Therefore, the production rate *g*SiO*<sup>x</sup>* of SiO*<sup>x</sup>* in Eq. (2) is assumed to be almost equal to the production rate *g*SiO of SiO.

*g*SiO at temperature *T* can be calculated as the ratio of the molar concentration *c*SiO of SiO at temperature *T* to the maximum value *c*<sup>0</sup> SiO. This *<sup>c</sup>*<sup>0</sup> SiO is the molar concentration when SiO2 changes to SiO via Eq. (1) with a yield of about 100%.

We assumed that the pyrolysis reaction system given by Eq. (1) reaches its equilibrium state during fiber fuse propagation. The equilibrium constant for Eq. (1) is denoted as *Kc*. The value of *c*SiO at temperature *T* was calculated using *Kc* as described below.

First, the initial molar concentration *c<sup>s</sup>* SiO2 (= 0.0366 mol cm<sup>−</sup>3) is denoted by *<sup>a</sup>*. We consider the case that the SiO2 concentration decreases to *a* − *y* via the pyrolysis reaction of Eq. (1). In this case, the molar concentration of SiO and the molar concentration *c*O2 of O2 are expressed as *y* and *y*/2, respectively. *Kc* is given in terms of *a* and *y* as follows:

$$K\_{\mathbb{C}} = \frac{c\_{\text{SiO}}c\_{\text{O}\_2}^{1/2}}{c\_{\text{SiO}\_2}} = \frac{y(y/2)^{1/2}}{a-y}.\tag{3}$$

Rearranging Eq. (3), we obtain a cubic equation in *y*. The solution of this equation is given by

$$y = \sqrt[3]{\text{C}} + \frac{4K\_c^2}{3\sqrt[3]{\text{C}}} \left(\frac{K\_c^2}{3} - a\right) + \frac{2K\_c^2}{3},\tag{4}$$

where

6 Optical Fiber

fSi

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

**2.2. Production rate of SiO***<sup>x</sup>*

temperature *T* to the maximum value *c*<sup>0</sup>

First, the initial molar concentration *c<sup>s</sup>*

changes to SiO via Eq. (1) with a yield of about 100%.

rate *g*SiO of SiO.

by

0.0

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

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 x

The most thermally stable compound in SiO*x* generated via Eq. (1) is SiO (*x* = 1). Therefore, the production rate *g*SiO*<sup>x</sup>* of SiO*<sup>x</sup>* in Eq. (2) is assumed to be almost equal to the production

*g*SiO at temperature *T* can be calculated as the ratio of the molar concentration *c*SiO of SiO at

We assumed that the pyrolysis reaction system given by Eq. (1) reaches its equilibrium state during fiber fuse propagation. The equilibrium constant for Eq. (1) is denoted as *Kc*. The

the case that the SiO2 concentration decreases to *a* − *y* via the pyrolysis reaction of Eq. (1). In this case, the molar concentration of SiO and the molar concentration *c*O2 of O2 are expressed

> 1/2 O2 *c*SiO2

Rearranging Eq. (3), we obtain a cubic equation in *y*. The solution of this equation is given

 *K*<sup>2</sup> *c* <sup>3</sup> <sup>−</sup> *<sup>a</sup>* + 2*K*<sup>2</sup> *c*

4*K*<sup>2</sup> *c* 3 √3 *C* <sup>=</sup> *<sup>y</sup>*(*y*/2)

1/2

value of *c*SiO at temperature *T* was calculated using *Kc* as described below.

*Kc* <sup>=</sup> *<sup>c</sup>*SiO*<sup>c</sup>*

as *y* and *y*/2, respectively. *Kc* is given in terms of *a* and *y* as follows:

*<sup>y</sup>* <sup>=</sup> <sup>√</sup><sup>3</sup> *<sup>C</sup>* <sup>+</sup>

SiO. This *<sup>c</sup>*<sup>0</sup>

SiOx

*<sup>α</sup>*SiO*<sup>x</sup>* (*T*) = *<sup>g</sup>*SiO*<sup>x</sup>* (*T*)*f*Si(*x*)*α*Si(*T*). (2)

SiO is the molar concentration when SiO2

*<sup>a</sup>* <sup>−</sup> *<sup>y</sup>* . (3)

<sup>3</sup> , (4)

SiO2 (= 0.0366 mol cm<sup>−</sup>3) is denoted by *<sup>a</sup>*. We consider

0.2

0.4

0.6

0.8

1.0

$$\begin{split} C &= a^2 K\_c^2 - \frac{4}{3} a K\_c^4 + \frac{8}{27} K\_c^6 \\ &+ a^2 K\_c^2 \sqrt{1 - \frac{8 K\_c^2}{27a}}. \end{split} \tag{5}$$

It is well known that the equilibrium constant *Kc* is related to the standard Gibbs energy change ∆*rG*<sup>0</sup> for Eq. (1). The relationship between *Kc* and ∆*rG*<sup>0</sup> is given by [75]

$$
\ln K\_c = \frac{-\Delta\_r G^0}{RT} \,, \tag{6}
$$

where *R* is the gas constant. The ∆*rG*<sup>0</sup> value for Eq. (1) is given by

$$
\begin{split}
\Delta\_{\rm f}G^{0} &= \Delta\_{f}G\_{\rm SiO}^{0} + (1/2)\Delta\_{f}G\_{\rm O\_{2}}^{0} \\ &- \Delta\_{f}G\_{\rm SiO\_{2}}^{0}
\end{split}
\tag{7}
$$

where <sup>∆</sup>*<sup>f</sup> <sup>G</sup>*<sup>0</sup> is the standard production Gibbs energy of a reactant and/or a product.

Vitreous silica (SiO2) is a solid at the standard temperature (298.15 K). It melts at high temeperatures of above 1,996 K, and becomes a liquid. It also becomes a vapor at temperatures of above 3,000 K.

The standard production Gibbs energies <sup>∆</sup>*<sup>f</sup> <sup>G</sup>*<sup>0</sup> SiO2 , <sup>∆</sup>*<sup>f</sup> <sup>G</sup>*<sup>0</sup> SiO, and <sup>∆</sup>*<sup>f</sup> <sup>G</sup>*<sup>0</sup> O2 in each phase have been published [76]. Thus, using these <sup>∆</sup>*<sup>f</sup> <sup>G</sup>*<sup>0</sup> values, we first calculated the standard Gibbs energy change ∆*rG*<sup>0</sup> for Eq. (1). Next, we calculated *Kc* by substituting the ∆*rG*<sup>0</sup> value into Eq. (6). In this way, we computed *c*SiO (= *y*) at temperature *T* by substituting *Kc* and *a* (= 0.0366 mol cm<sup>−</sup>3) into Eqs. (4) and (5). The relationship between *<sup>c</sup>*SiO and *<sup>T</sup>* is shown in Figure 5. *c*SiO increases with increasing *T* and gradually approaches its maximum value (*c*0 SiO <sup>∼</sup><sup>=</sup> 0.0366 mol cm<sup>−</sup>3) at *<sup>T</sup>* of about 3,200 K.

Therefore, *g*SiO*<sup>x</sup>* at temperature *T* was estimated by dividing the value of *c*SiO calculated above by *c*<sup>0</sup> SiO (∼<sup>=</sup> 0.0366 mol cm<sup>−</sup>3).

#### **2.3. Absorption coefficient of amorphous Si**

The optical absorption spectrum of amorphous Si was reported by Brodsky *et al.* [77]. The absorption coefficient *α*Si of amorphous Si in Eq. (2) was estimated as follows.

**Figure 5.** Molar concentration of SiO vs. temperature.

First, we consider the absorption coefficient *α*Si of an interband transition region, where the photon energy ¯*hω* is larger than the energy gap *Eg* (= 1.26 eV) of amorphous Si. In this case, the values of *<sup>α</sup>*Si (cm−<sup>1</sup> unit) near *Eg* is given by

$$\alpha\_{\rm Si}(\omega) = B \frac{\left(\hbar \omega - E\_{\rm g}\right)^2}{\hbar \omega},\tag{8}$$

where the parameter *<sup>B</sup>* is 5.06 <sup>×</sup> <sup>10</sup><sup>5</sup> cm−<sup>1</sup> eV<sup>−</sup>1. The *<sup>α</sup>*Si values calculated by Eq. (8) are in good agreement with the experimental values reported by Brodsky *et al.* [77] for the case of *h*¯ *ω* > 1.5 eV.

Next, we consider the values of *<sup>α</sup>*Si in the low-energy region, where ¯*h<sup>ω</sup>* <sup>&</sup>lt; *Eg*. In this region, an absorption edge spectrum is broadened as a result of the interaction between optical phonons and electrons (or excitons). *α*Si in this region exhibits exponential behavior (referred to as an"exponential tail" or "Urbach tail") as follows [78]:

$$\mathfrak{a}\_{\rm Si}(\omega) = \mathfrak{a}\_0 \exp\left[\frac{\gamma(\hbar\omega - E\_{\rm g})}{kT^\*}\right],\tag{9}$$

where *<sup>α</sup>*<sup>0</sup> and *<sup>γ</sup>* are parameters and *<sup>k</sup>* is the Boltzmann constant. *<sup>T</sup>*<sup>∗</sup> is the effective temperature. If the characteristic temperature of phonons is denoted by *θ*, then *T*<sup>∗</sup> is given by [79]

$$T^\* = \frac{\theta}{2}\coth\left(\frac{\theta}{2T}\right). \tag{10}$$

For Si, *<sup>θ</sup>* = 600 K [80]. *<sup>α</sup>*<sup>0</sup> and *<sup>γ</sup>* are estimated to be 9.016 × <sup>10</sup><sup>3</sup> cm−<sup>1</sup> and 0.14, respectively, by simulation using the experimental *α*Si values reported by Brodsky *et al.* [77].

It is known that the energy gap *Eg* of Si decreases linearly with increasing *T* at *T* > 200 K [81]. The temperature dependence of *Eg* (eV unit) at *T* > 300 K is given by

$$E\_{\mathcal{S}} = 1.26 - \beta T\_{\prime} \tag{11}$$

where *β* is the temperature coefficient of the energy gap and takes a value of 4.2 × 10−<sup>4</sup> eV/K [82].

## **2.4. Absorption coefficient of SiO***<sup>x</sup>*

8 Optical Fiber

*h*¯ *ω* > 1.5 eV.

by [79]


Log

**Figure 5.** Molar concentration of SiO vs. temperature.

the values of *<sup>α</sup>*Si (cm−<sup>1</sup> unit) near *Eg* is given by

cSiO [

mol cm-3 ] c0 SiO

1000 2000 3000 4000 5000 Temperature T (K)

First, we consider the absorption coefficient *α*Si of an interband transition region, where the photon energy ¯*hω* is larger than the energy gap *Eg* (= 1.26 eV) of amorphous Si. In this case,

*<sup>α</sup>*Si(*ω*) = *<sup>B</sup>* (*h*¯ *<sup>ω</sup>* <sup>−</sup> *Eg*)

where the parameter *<sup>B</sup>* is 5.06 <sup>×</sup> <sup>10</sup><sup>5</sup> cm−<sup>1</sup> eV<sup>−</sup>1. The *<sup>α</sup>*Si values calculated by Eq. (8) are in good agreement with the experimental values reported by Brodsky *et al.* [77] for the case of

Next, we consider the values of *<sup>α</sup>*Si in the low-energy region, where ¯*h<sup>ω</sup>* <sup>&</sup>lt; *Eg*. In this region, an absorption edge spectrum is broadened as a result of the interaction between optical phonons and electrons (or excitons). *α*Si in this region exhibits exponential behavior

where *<sup>α</sup>*<sup>0</sup> and *<sup>γ</sup>* are parameters and *<sup>k</sup>* is the Boltzmann constant. *<sup>T</sup>*<sup>∗</sup> is the effective temperature. If the characteristic temperature of phonons is denoted by *θ*, then *T*<sup>∗</sup> is given

<sup>2</sup> coth

For Si, *<sup>θ</sup>* = 600 K [80]. *<sup>α</sup>*<sup>0</sup> and *<sup>γ</sup>* are estimated to be 9.016 × <sup>10</sup><sup>3</sup> cm−<sup>1</sup> and 0.14, respectively,

 *θ* 2*T* 

 *γ*(*h*¯ *ω* − *Eg*) *kT*<sup>∗</sup>

(referred to as an"exponential tail" or "Urbach tail") as follows [78]:

*<sup>α</sup>*Si(*ω*) = *<sup>α</sup>*<sup>0</sup> exp

*<sup>T</sup>*<sup>∗</sup> <sup>=</sup> *<sup>θ</sup>*

by simulation using the experimental *α*Si values reported by Brodsky *et al.* [77].

2

*<sup>h</sup>*¯ *<sup>ω</sup>* , (8)

, (9)

. (10)

On the basis of the above results, the temperature dependence of *α*SiO*<sup>x</sup>* at 1.064 *µ*m (¯*hω* = 1.17 eV) was calculated using Eq. (2) and the *c*SiO values shown in Figure 5. *f*Si = 0.0625 (*x* = 1) was used in the calculation. The results are shown in Figure 6.

**Figure 6.** Absorption coefficient of SiO*<sup>x</sup>* at 1.064 *µ*m vs. temperature. The thick solid line was calculated using Eq. (2). The dotted line shows the values estimated by Kashyap *et al.*

The *<sup>α</sup>*SiO*<sup>x</sup>* values (shown as the thick solid line in Figure 6) are about 1.5 <sup>×</sup> 104 <sup>m</sup>−<sup>1</sup> at 2,800 K and about 3.5 × 104 m−<sup>1</sup> at 2,950 K. These values are close to 4 × 104 m−<sup>1</sup> (shown as the dotted line in Figure 6), which was estimated by simulation using the finite element method [19].

Moreover, it turns out that *<sup>α</sup>*SiO*<sup>x</sup>* reaches its maxium value (about 9.47 <sup>×</sup> 104 <sup>m</sup><sup>−</sup>1) at about 3,150 K, then decreases gradually with increasing temperature.

Compared with the stepwise shape (dotted line in Figure 6) of the absorption coefficient assumed by Kashyap *et al.*, the maximum value (about 9.47 <sup>×</sup> <sup>10</sup><sup>4</sup> <sup>m</sup><sup>−</sup>1) of *<sup>α</sup>*SiO*<sup>x</sup>* is about 2.4 times the assumed value (4 × 104 m<sup>−</sup>1), and the graph has the shape of a somewhat distorted echelon form.

As mentioned above, SiO*x* is produced in the pyrolysis of vitreous silica, and it induces a very large amount of optical absorption with a large absorption coefficient of 10<sup>4</sup> m−<sup>1</sup> order at high temperatures of above 2,800 K. It is thought that the large amount of optical absorption at high temperatures is the cause of the genesis of the fiber fuse phenomenon.

We investigated the thermal conduction behavior within an SMF by numerical computation using the thermochemical SiO*x* production model. In the next section, we describe the results of some numerical calculations related to the thermal conduction process in an SMF.

## **3. Simulation of fiber fuse in SMF**

#### **3.1. Heat conduction in SMF**

We assume the SMF 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 length ∆*L* is heated to a temperature of *T*0 *<sup>c</sup>* (> *Ta*) (see Figure 7). Such a region, called the "hot zone" in Figure 7, can be generated by heating the fiber end faces using the arc discharge of a fusion splice machine.

**Figure 7.** Hot zone in the core layer.

As explained above, the optical absorption coefficient *α* of the core layer in an optical fiber is a function of temperature *T*, and *α* increases with increasing *T*. In the hot zone in Figure 7, the *α* values are larger than those in the remainder of the core region because of its elevated temperature. Thus, as light propagates along the positive direction (away from the light source) in this zone, considerable 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 SMF is given by [83]

$$\begin{split} \rho \mathbb{C}\_p \frac{\partial T}{\partial t} &= \lambda \left( \frac{\partial^2 T}{\partial r^2} + \frac{1}{r} \frac{\partial T}{\partial r} + \frac{\partial^2 T}{\partial z^2} \right) \\ &+ \dot{Q} \end{split} \tag{12}$$

where *ρ*, *Cp*, and *λ* are the density, specific heat, and thermal conductivity of the fiber, respectively. The values of *ρ*, *Cp*, and *λ* used for the calculation are described in the next subsection.

The last term *Q*˙ in Eq. (12) represents the heat source resulting from light absorption, which is only required for the hot zone in the fiber core. *Q*˙ can be expressed by

$$
\dot{Q} = \mathfrak{a}I,\tag{13}
$$

where *I* is the optical power intensity in the core layer, which can be estimated by dividing the incident optical power *P* by the effective area *Aeff* of the fiber.

#### **3.2. Heat conduction parameters**

10 Optical Fiber

*T*0

high temperatures of above 2,800 K. It is thought that the large amount of optical absorption

We investigated the thermal conduction behavior within an SMF by numerical computation using the thermochemical SiO*x* production model. In the next section, we describe the results

We assume the SMF 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 length ∆*L* is heated to a temperature of

*<sup>c</sup>* (> *Ta*) (see Figure 7). Such a region, called the "hot zone" in Figure 7, can be generated

∆L

Ta

As explained above, the optical absorption coefficient *α* of the core layer in an optical fiber is a function of temperature *T*, and *α* increases with increasing *T*. In the hot zone in Figure 7, the *α* values are larger than those in the remainder of the core region because of its elevated temperature. Thus, as light propagates along the positive direction (away from the light

In the case of a heat source in part of the core layer, the nonsteady heat conduction equation

where *ρ*, *Cp*, and *λ* are the density, specific heat, and thermal conductivity of the fiber, respectively. The values of *ρ*, *Cp*, and *λ* used for the calculation are described in the next

The last term *Q*˙ in Eq. (12) represents the heat source resulting from light absorption, which

1 *r ∂T ∂r* + *∂*2*T ∂z*<sup>2</sup> 

 *∂*2*T <sup>∂</sup>r*<sup>2</sup> <sup>+</sup>

source) in this zone, considerable heat is produced by light absorption.

is only required for the hot zone in the fiber core. *Q*˙ can be expressed by

for the temperature field *T*(*r*, *z*, *t*) in the SMF is given by [83]

*ρCp ∂T <sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>λ</sup>* Tc 0

rf

+ *Q*˙ , (12)

of some numerical calculations related to the thermal conduction process in an SMF.

by heating the fiber end faces using the arc discharge of a fusion splice machine.

at high temperatures is the cause of the genesis of the fiber fuse phenomenon.

**3. Simulation of fiber fuse in SMF**

P0

**3.1. Heat conduction in SMF**

**Figure 7.** Hot zone in the core layer.

subsection.

In the heat conduction calculation for SMFs, we used the following values of *λ* (W m−<sup>1</sup> K<sup>−</sup>1), *ρ* (kg m<sup>−</sup>3), and *Cp* (J kg−<sup>1</sup> K<sup>−</sup>1) in each temperature range. The unit of *T* is K (Kelvin).

(1) Parameters in the temperature range from room temperature (298 K) to 1,996 K [84]:

$$\begin{aligned} C\_p &= 1194.564 + 31.541 \times 10^{-3}T \\ &- 651.396 \times 10^5 T^{-2} \\ \lambda &= 9.2 \\ \rho &= 2,200.0. \end{aligned}$$

(2) Parameters in the temperature range from 1,996 to 3,000 K [84]:

$$\begin{aligned} \mathsf{C}\_p &= 1, 430.490 \\ \lambda &= 9.2 \\ \rho &= 2, 200.0. \end{aligned}$$

(3) Parameters for *T* > 3,000 K:

$$\begin{aligned} \mathbf{C}\_p &= 844.4 \\ \lambda &= 0.0025247 \sqrt{T} \\ &+ 1.84275 \times 10^{-13} T^{5/2} \\ \rho &= 2,024.0. \end{aligned}$$

In (3), the first term in the expression for *λ* was estimated by substituting the parameters for SiO in the following equation, which represents the thermal conductivity of diatomic molecules derived from the kinetic theory of gases [85]:

$$
\lambda = \frac{5}{3\sigma^2} \sqrt{\left(\frac{k}{\pi}\right)^3 \frac{N\_A}{M}} \sqrt{T} \,\tag{14}
$$

where *NA* is Avogadro's number, *M* (=44.0854) is the molecular weight of SiO, and *σ* (= 1.5 Å) is half of the collision diameter.

Moreover, in (3), the second term in the expression for *λ* was estimated by mutiplying the following thermal conductivity equation for weakly ionized gas plasma [86] by the correction factor of 1/20:

$$
\lambda = \frac{5N\_\text{c}k^2T}{2m\_\text{c}\nu\_\text{c}},
\tag{15}
$$

where *me* is the electron mass, *Ne* is the number density of electrons in the plasma, and *ν<sup>c</sup>* is the collision frequency. If we assume that electrons mainly collide with ions with a charge of +1 in the plasma, then *ν<sup>c</sup>* is given by [87]

$$\nu\_{\varepsilon} = \sqrt{\frac{2}{9\pi}} \frac{N\_{\varepsilon}e^{4}}{\varepsilon\_{0}^{2}m\_{fe}^{2}} \left(\frac{m\_{\varepsilon}}{3kT}\right)^{3/2} \ln \Lambda,\tag{16}$$

where *e* is the electron charge and *ε*<sup>0</sup> is the permittivity of free space. ln Λ is the so-called Coulomb logarithm, and it takes values of 4–34 in the electron temperature range of 102–108 K and the *Ne* range of 1–1024 cm−<sup>3</sup> [88]. In our calculation, we used ln Λ = 16.155, which corresponds to the case of *T* ∼ 104 K and *Ne* ∼ 10<sup>6</sup> cm<sup>−</sup>3.

#### **3.3. Boundary and initial conditions for heat conduction**

We solved Eq. (12) using the explicit finite-difference method (FDM) [89] under the boundary and initial conditions described below.

The area in the numerical calculation had 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 24 and 2,000 divisions in the *r* and *z* directions, respectively, and we set the calculation time interval to 1 *µ*s. We assumed that the hot zone was located at the center of the fiber (length 2*L*) and that the length ∆*L* of the hot zone was 40 *µ*m.

The boundary conditions are as follows:

(1) Since the temperature distribution of the optical fiber is axisymmetric, the amount of heat conducted per unit area (heat flux) in the *r* direction is set to 0 at the fiber center (*r* = 0) as follows:

$$\left. -\lambda \frac{\partial T}{\partial r} \right|\_{r=0} = 0.\tag{17}$$

(2) At the outer fiber surface (*r* = *rf* ), the amount of heat conducted per unit area (heat flux) is dissipated by radiative transfer or heat transfer to the open air and/or the jacketing layer (*T* = *Ta*) as follows:

$$-\lambda \frac{\partial T}{\partial r}\bigg|\_{r=r\_f} = \sigma\_S \epsilon\_\varepsilon \left(T^4 - T\_a^4\right)$$

$$+\frac{\lambda}{\delta r\_t} \left(T - T\_a\right) . \tag{18}$$

where *σ<sup>S</sup>* is the Stefan-Boltzmann constant and *ǫ<sup>e</sup>* (∼ 0.9) is the emissivity of the surface. *δrt* is the thickness of the thermal boundary layer and *δrt* = *δr* is assumed in our calculation, where *δr* is the step size along the *r* axis.

12 Optical Fiber

factor of 1/20:

+1 in the plasma, then *ν<sup>c</sup>* is given by [87]

and initial conditions described below.

The boundary conditions are as follows:

the hot zone was 40 *µ*m.

(*T* = *Ta*) as follows:

follows:

*ν<sup>c</sup>* =

corresponds to the case of *T* ∼ 104 K and *Ne* ∼ 10<sup>6</sup> cm<sup>−</sup>3.

**3.3. Boundary and initial conditions for heat conduction**

 2 9*π*

Moreover, in (3), the second term in the expression for *λ* was estimated by mutiplying the following thermal conductivity equation for weakly ionized gas plasma [86] by the correction

> *<sup>λ</sup>* <sup>=</sup> <sup>5</sup>*Nek*2*<sup>T</sup>* 2*meν<sup>c</sup>*

where *me* is the electron mass, *Ne* is the number density of electrons in the plasma, and *ν<sup>c</sup>* is the collision frequency. If we assume that electrons mainly collide with ions with a charge of

> *Nee*<sup>4</sup> *ε*2 0*m*<sup>2</sup> *f e*

where *e* is the electron charge and *ε*<sup>0</sup> is the permittivity of free space. ln Λ is the so-called Coulomb logarithm, and it takes values of 4–34 in the electron temperature range of 102–108 K and the *Ne* range of 1–1024 cm−<sup>3</sup> [88]. In our calculation, we used ln Λ = 16.155, which

We solved Eq. (12) using the explicit finite-difference method (FDM) [89] under the boundary

The area in the numerical calculation had 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 24 and 2,000 divisions in the *r* and *z* directions, respectively, and we set the calculation time interval to 1 *µ*s. We assumed that the hot zone was located at the center of the fiber (length 2*L*) and that the length ∆*L* of

(1) Since the temperature distribution of the optical fiber is axisymmetric, the amount of heat conducted per unit area (heat flux) in the *r* direction is set to 0 at the fiber center (*r* = 0) as

(2) At the outer fiber surface (*r* = *rf* ), the amount of heat conducted per unit area (heat flux) is dissipated by radiative transfer or heat transfer to the open air and/or the jacketing layer

> = *σSǫ<sup>e</sup> T*<sup>4</sup> − *T*<sup>4</sup> *a*

+ *λ δrt*

<sup>−</sup> *<sup>λ</sup> <sup>∂</sup><sup>T</sup> ∂r r*=0

<sup>−</sup> *<sup>λ</sup> <sup>∂</sup><sup>T</sup> ∂r r*=*rf*  *me* 3*kT*

3/2

, (15)

ln Λ, (16)

= 0. (17)

(*T* − *Ta*), (18)

(3) At the center (*z* = 0) of the hot zone, the temperature of the fiber core center is *T*<sup>0</sup> *c* . Also, there is heat inflow along the *z* axis only at the core center (*r* = 0), which is attributable to light absorption. Moreover, when *r* �= 0 and *z* = 0, there is neither heat inflow nor heat outflow along the *z* axis. These conditions are given by

$$-\lambda \frac{\partial T}{\partial z}\bigg|\_{z=0} = \begin{cases} \alpha I, & \text{if } r = 0\\ 0 & \text{if } r \neq 0. \end{cases} \tag{19}$$

(4) At both fiber ends (*z* = ±*L*), the amount of heat conducted per unit area (heat flux) is dissipated by radiation transfer to the open air (*T* = *Ta*) as follows:

$$\left. -\lambda \frac{\partial T}{\partial z} \right|\_{z=\pm L} = \sigma\_S \epsilon\_\varepsilon \left( T^4 - T\_a^4 \right). \tag{20}$$

In contrast, as initial conditions, *T* = *Ta* in the optical fiber at *t* = 0, except in the hot zone, and the core-center temperature in the hot zone is equal to *T*<sup>0</sup> *<sup>c</sup>* (> *Ta*) as follows:

$$T(0, z, 0) = \begin{cases} T\_{a\prime} & \text{if } -L \le z < -\Delta L/2 \\ T\_{c\prime}^0 & \text{if } -\Delta L/2 \le z \le \Delta L/2 \\ T\_a & \text{if } \Delta L/2 < z \le L. \end{cases} \tag{21}$$

When light propagates through the fiber core (*r* = 0) along the *z* direction (away from the light source), the incident laser power *P* decreases because of the nonzero optical absorption coefficient *α*. When the laser light propagates from *z* to *z* + *δz* along the *z* axis at *r* = 0, the *P* value is given by

$$P = P\_0 \exp\left(-a\delta z - \int\_{-L}^{z} a(T)dz\right),\tag{22}$$

where *P*<sup>0</sup> is the initial laser power. The second term on the right-hand side expresses the optical absorption loss when the light propagates through a distance of *z* − *L*.

The results described above assume that the laser light propagates through the fiber core along the positive *z* direction (away from the light source).

When the core layer is heated to above the vaporization point of silica (∼ 3273 K), an enclosed hollow cavity is produced in the core center. This cavity contains oxygen, which is produced by the pyrolysis reaction of Eq. (1). The heat conductivity *κ* of the oxygen (0.03 W m−<sup>1</sup> K<sup>−</sup>1) is two orders smaller than that of the silicate glass. Therefore, the heat transferred in the silica core is stopped at the cavity.

Moreover, as the cavity has a refractive index of *n* ∼ 1, which is smaller than that of the silica core (*n*<sup>1</sup> = 1.46), the light propagating in the core layer is reflected at the cavity wall. When the light direction is reversed at the cavity, the heat source term *αIr* resulting from the optical absorption of the reflected light is added to *αI* in Eq. (13), where *Ir* is the optical power intensity of the reflected laser light.

We consider the *P* value at a *z* position located near the cavity wall. The length of this position is assumed to be *δz*. The laser light reaches *z*, propagates through a distance of *δz*, and then reaches the cavity wall, whose coordinate is *zv*. Next, the light is reflected at the cavity wall and propagates in the negative *z* direction, and then reaches *z* again. In such a case, the *P* value at *z* is given by

$$P = P\_0 \exp\left(-\alpha \delta z - \int\_{-L}^{z} \alpha(T) dz\right)$$

$$\times \operatorname{Rexp}\left(-2 \int\_{z+\delta z}^{z\_0} \alpha(T) dz\right),\tag{23}$$

where *R* is the reflectivity at the boundary of the silica core and the cavity. *R* is given by

$$R = \left(\frac{n\_1 - 1}{n\_1 + 1}\right)^2. \tag{24}$$

The second integral on the right-hand side of Eq. (23), which is related to the reflected light, slightly affects the occurrence of the fiber fuse. However, by taking this term into consideration, the calculated fiber fuse velocities fit the experimental values. For this reason, in the present work, we decided to take into account the effect of the reflected light.

In the following section, we describe the calculated time (*t*) dependence of *T*(*r*, *z*) in an SMF.

#### **3.4. Propagation of fiber fuse in SMF**

In the calculation, we used *T*<sup>0</sup> *<sup>c</sup>* = 2923 K and *Ta* = 298 K. It was assumed that laser light of wavelength *λ*<sup>0</sup> = 1.064 *µ*m and *P*<sup>0</sup> = 2 W was incident to an SMF-28 optical fiber, which has a core diameter of 2 *rc* = 8.2 *µ*m, a refractive index difference of ∆ = 0.36%, and *Aeff* = 49.4091 × 10−<sup>12</sup> m2.

We calculated the *T*(*r*, *z*) values at *t* = 1, 11, and 21 ms after the incidence of the 2 W laser light. The calculated results are shown in Figures 8 ∼ 10, respectively.

As shown in Figure 8, the core center temperature near the end of the hot zone (*z* = -0.7 mm) changes abruptly to a large value of about 3.4 ×104 K after 1 ms. This rapid rise in the temperature initiates the fiber fuse phenomenon as shown in Figures 9 and 10. After 11 and 21 ms, the high-temperature front in the core layer reached *z* values of -6.1 and -11.5 mm, respectively. The average propagation velocity *vf* was estimated to be 0.54 m/s using these data.

**Figure 8.** Temperature field in SMF-28 after 1 ms when *P*<sup>0</sup> = 2 W at *λ*<sup>0</sup> = 1.064 *µ*m.

14 Optical Fiber

intensity of the reflected laser light.

case, the *P* value at *z* is given by

**3.4. Propagation of fiber fuse in SMF**

In the calculation, we used *T*<sup>0</sup>

× 10−<sup>12</sup> m2.

data.

Moreover, as the cavity has a refractive index of *n* ∼ 1, which is smaller than that of the silica core (*n*<sup>1</sup> = 1.46), the light propagating in the core layer is reflected at the cavity wall. When the light direction is reversed at the cavity, the heat source term *αIr* resulting from the optical absorption of the reflected light is added to *αI* in Eq. (13), where *Ir* is the optical power

We consider the *P* value at a *z* position located near the cavity wall. The length of this position is assumed to be *δz*. The laser light reaches *z*, propagates through a distance of *δz*, and then reaches the cavity wall, whose coordinate is *zv*. Next, the light is reflected at the cavity wall and propagates in the negative *z* direction, and then reaches *z* again. In such a

−*αδz* −

where *R* is the reflectivity at the boundary of the silica core and the cavity. *R* is given by

 *n*<sup>1</sup> − 1 *n*<sup>1</sup> + 1

The second integral on the right-hand side of Eq. (23), which is related to the reflected light, slightly affects the occurrence of the fiber fuse. However, by taking this term into consideration, the calculated fiber fuse velocities fit the experimental values. For this reason,

In the following section, we describe the calculated time (*t*) dependence of *T*(*r*, *z*) in an SMF.

wavelength *λ*<sup>0</sup> = 1.064 *µ*m and *P*<sup>0</sup> = 2 W was incident to an SMF-28 optical fiber, which has a core diameter of 2 *rc* = 8.2 *µ*m, a refractive index difference of ∆ = 0.36%, and *Aeff* = 49.4091

We calculated the *T*(*r*, *z*) values at *t* = 1, 11, and 21 ms after the incidence of the 2 W laser

As shown in Figure 8, the core center temperature near the end of the hot zone (*z* = -0.7 mm) changes abruptly to a large value of about 3.4 ×104 K after 1 ms. This rapid rise in the temperature initiates the fiber fuse phenomenon as shown in Figures 9 and 10. After 11 and 21 ms, the high-temperature front in the core layer reached *z* values of -6.1 and -11.5 mm, respectively. The average propagation velocity *vf* was estimated to be 0.54 m/s using these

light. The calculated results are shown in Figures 8 ∼ 10, respectively.

 *z* −*L*

*α*(*T*)*dz* 

2

*α*(*T*)*dz* 

*<sup>c</sup>* = 2923 K and *Ta* = 298 K. It was assumed that laser light of

, (23)

. (24)

*P* = *P*<sup>0</sup> exp

× *R* exp

 −2 *zv z*+*δz*

*R* =

in the present work, we decided to take into account the effect of the reflected light.

**Figure 9.** Temperature field in SMF-28 after 11 ms when *P*<sup>0</sup> = 2 W at *λ*<sup>0</sup> = 1.064 *µ*m.

**Figure 10.** Temperature field in SMF-28 after 21 ms when *P*<sup>0</sup> = 2 W at *λ*<sup>0</sup> = 1.064 *µ*m.

When the laser light of *λ*<sup>0</sup> = 1.064 *µ*m and *P*<sup>0</sup> = 2 W is incident to the SMF-28 optical fiber, the optical power intensity is *I* = 4.048 MW/cm2. The fiber fuse velocity at this value of *I* is estimated to be about 0.55 m/s (see Figure 10 in [69]). This value is in very good agreement with the upper value obtained by calculation (0.54 m/s).

On the other hand, Hand and Russell measured the fiber fuse temperature to be 5,400 K [5], and Dianov *et al.* obtained a temperature of 4,700–10,500 K [28],[90] by measurement. They estimated the temperatures from precisely measured spectral data in the 600–1,400 nm [5] and 500–800 nm [28],[90] regions, while assuming blackbody radiation.

In our calculation, the temperature distribution of the fiber fuse in the core center is shown in Figure 11. Similar shapes of temperature distribution were reported by Kashyap *et al.* [19]

**Figure 11.** Temperature distribution of the SMF-28 core center vs. length along the *z* direction when *P*<sup>0</sup> = 2 W at *λ*<sup>0</sup> = 1.064 *µ*m. The time after the laser incidence is 10 ms.

and Facão *et al.* [56].

As shown in Figure 11, it is clear that the sharp temperature peak is located near the light source, and a relatively high temperature plateau of about 5,000 K extends over about 1.5 mm behind the sharp peak. This region, called the "radiation zone" in Figure 11, exhibits high temperatures of above 323 K.

When gaseous SiO and/or SiO2 molecules are heated to high temperatures above 5,000 K, they decompose to form Si and O atoms, and finally becomes Si<sup>+</sup> and O<sup>+</sup> ions and electrons in the ionized gas plasma state [91]. The numbers of electrons and ions in the plasma front, which exhibits sharp temperature peak, are larger than those in the plateau region. However, as the plasma tends to restore electrical neutrality, the motions of the electrons and ions will not produce any change on the initial temperature distribution shown in Figure 11, except for the peak temperature reduction of the plasma front due to energy loss induced by electron-ion collisions.

In the ionized gas plasma, electron-ion collisions generate electromagnatic radiation because of the deceleration during the collisions. This bremsstrahlung emission [92], [93] is a universal and irreducible process of energy loss. If we assume that electrons mainly collide with ions with a charge of +1 in the plasma, the spectral radiance function *Ip* for the bremsstrahlung emission is given by [94]

$$I\_p = \frac{N\_c^2 \nu^2}{\sqrt{T}c^2} \exp\left(-\frac{h\nu}{kT}\right),\tag{25}$$

where *Ne* is the number density of electrons in the plasma, *h* is Planck's constant, and *ν* is the optical frequency. This functin is directly proportionate to *ν*<sup>2</sup> in the case of *hν* ≪ *kT*.

On the other hand, in the case of *hν* ≪ *kT*, the spectral radiance function *Ib* for blackbody radiation is given by [95]

$$I\_b = \frac{2\pi\nu^2}{c^2} kT.\tag{26}$$

This is well-known as the Rayleigh-Jeans formula, and this function is proportional to *ν*2, too. Therefore, we assumed that the radiation zone, in which the bremsstrahlung emission of the plasma is liberated, can be treated as a blackbody because of its similar dependence on *ν*.

If we consider the radiation zone as a blackbody, that is isolated from the surrounding nonheated regions, it is expected that the radiation zone will exhibit a radiation spectrum with a broad range of optical frequencies ranging from ultraviolet to infrared. The blackbody temerature *Tb* of the zone is related to the frequency *ν<sup>m</sup>* of the spectral peak as follows [95]:

$$\nu\_m = \frac{2.82kT\_b}{h}.\tag{27}$$

If *Tav* is defined as the average temperature of the radiation zone, the *Tb* will be close to the *Tav*.

The relationship between the *Tav* value and the time after fiber fuse generation is shown in Figure 12. *Tav* exceeds 10000 K immediately after fiber fuse generation but is less than 7000 K after 4 ms. Then it gradually approaches about 5,700 K. The value of 5,760 K shown in Figure 12 is the average *Tav* value from 4 to 30 ms. This temperature (5,760 K) is close to the reported temperatures of 5,400 K [5] and about 5,800 K [90].

Thus, it was found that if a fiber fuse, which exhibits a sharp temperature peak located near the light source, is approximated as a blackbody isolated from the surrounding nonheated regions, its average temperature from 4 ms after the generation of the fiber fuse approaches the experimentally estimated radiation temperatures.

### **4. Modeling of Hole-Assisted Fiber**

16 Optical Fiber

When the laser light of *λ*<sup>0</sup> = 1.064 *µ*m and *P*<sup>0</sup> = 2 W is incident to the SMF-28 optical fiber, the optical power intensity is *I* = 4.048 MW/cm2. The fiber fuse velocity at this value of *I* is estimated to be about 0.55 m/s (see Figure 10 in [69]). This value is in very good agreement

On the other hand, Hand and Russell measured the fiber fuse temperature to be 5,400 K [5], and Dianov *et al.* obtained a temperature of 4,700–10,500 K [28],[90] by measurement. They estimated the temperatures from precisely measured spectral data in the 600–1,400 nm [5]

In our calculation, the temperature distribution of the fiber fuse in the core center is shown in Figure 11. Similar shapes of temperature distribution were reported by Kashyap *et al.* [19]


z (mm)

**Figure 11.** Temperature distribution of the SMF-28 core center vs. length along the *z* direction when *P*<sup>0</sup> = 2 W at *λ*<sup>0</sup> = 1.064

As shown in Figure 11, it is clear that the sharp temperature peak is located near the light source, and a relatively high temperature plateau of about 5,000 K extends over about 1.5 mm behind the sharp peak. This region, called the "radiation zone" in Figure 11, exhibits

When gaseous SiO and/or SiO2 molecules are heated to high temperatures above 5,000 K, they decompose to form Si and O atoms, and finally becomes Si<sup>+</sup> and O<sup>+</sup> ions and electrons in the ionized gas plasma state [91]. The numbers of electrons and ions in the plasma front, which exhibits sharp temperature peak, are larger than those in the plateau region. However, as the plasma tends to restore electrical neutrality, the motions of the electrons and ions will not produce any change on the initial temperature distribution shown in Figure 11, except for the peak temperature reduction of the plasma front due to energy loss induced by

In the ionized gas plasma, electron-ion collisions generate electromagnatic radiation because of the deceleration during the collisions. This bremsstrahlung emission [92], [93] is a universal and irreducible process of energy loss. If we assume that electrons mainly collide

Radiation Zone

with the upper value obtained by calculation (0.54 m/s).

0

10

20

T (x103K)

*µ*m. The time after the laser incidence is 10 ms.

high temperatures of above 323 K.

and Facão *et al.* [56].

electron-ion collisions.

30

40

and 500–800 nm [28],[90] regions, while assuming blackbody radiation.

In a Hole-Assisted Fiber (HAF), some holes exist around the main core as shown in Figure 1. In this HAF, the minimum distance of the holes from the main core is only (*Dhole* − 2*rc*)/2. These holes are filled with air at the same temperature as the surrounding air (*Ta*). Therefore, at the inner surfaces of the holes, it can be expected that heat transfer or radiative transfer

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

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 in the core center of the HAF.

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

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

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, it was assumed that heat transfer and radiative transfer occurred at the inner surface of the hole layer in addition to heat conduction between the silica glass and the silica-air mixture.

When a certain quantity of heat per unit area (heat flux) is conducted from the heated core center to the end of the first cladding layer (*r* = *Dhole*/2) of an optical fiber, a considerable amount of the heat flux is transmitted through the hole layer, and the residual heat flux stagnates in the first cladding layer because the thermal conductivity *κ* of the hole layer is lower than that of silica glass.

This heat flux stagnating in the first cladding layer can be dissipated by radiative transfer and heat transfer between the outer surface of the first cladding layer and the air in the hole layer with a temperature of *Ta*.

It is noteworthy here that the total surface area of the 6-air holes existing in an actual HAF is larger than the outer surface area of the first cladding layer when *Rh* = 2 (see Table 1).

The heat transfer results from the convection of air, which is generated by warming the air near the outer surface of the first cladding layer. In this heat transfer, there is a region (called the "thermal boundary layer") near the surface of the first cladding layer where the temperature of the air changes rapidly from a high value to *Ta*.
