3. Effect of nonlinearity parameters on cavity patterns

The nonlinearity parameter γ characterizes the oscillation pattern. The oscillatory motion for ε ¼ 9, β ¼ 6:5, and γ ¼ 0 was shown in Figure 5, where the perturbed density ρ<sup>1</sup> is plotted as a function of time. It can be seen in Figure 5 that the oscillations consist of sudden transitions between compressed and rarefied regions, and the retention time τ<sup>r</sup> of the rarefied regions equals that of the compressed regions τc. The relationship between the period Φ (¼ τ<sup>r</sup> þ τc) and the interval Λ is given by Eq. (4), and the relationship between τ<sup>r</sup> and the length l of the cavity is

$$d = \mathfrak{r}\_r \mathcal{V}\_f. \tag{8}$$

The Λ and l values of the motion corresponding to ε ¼ 9, β ¼ 6:5, and γ ¼ 0 are estimated to be about 10.8 and 21.6 μm, respectively, using Eqs. (4) and (8) and Vf ¼ 1 m/s. That is, l=Λ ¼ 0:5 in the case of γ ¼ 0.

Next, the oscillatory motion for γ ¼ 2 and � 2 with ε ¼ 9 and β ¼ 6:5 was examined. The calculated results are shown in Figures 10 and 11, respectively. As shown in Figure 10, the retention time τ<sup>r</sup> of the rarefied regions is larger than that of the compressed regions τc. As a result, the ratio l=Λ is larger than 0.5 in the case of

Figure 10. Time dependence of the perturbed density during fiber fuse propagation. ε ¼ 9, β ¼ 6:5, γ ¼ 2.

addition to Vf , Todoroki reported the P<sup>0</sup> dependence of Λ in an SMF-28e fiber at

mental Vf values [23, 26] and the calculated Φ values shown in Figure 8.

(Pth ≃1:3W [61]) to 9 W, Λð Þ P<sup>0</sup> can be represented by

Fiber Optics - From Fundamentals to Industrial Applications

The second term �<sup>ζ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Λð Þ¼ P<sup>0</sup> Φ0Vfð Þ P<sup>0</sup> 1 � ζ

represents the contribution of the nonlinearity to the overall Λ value.

In this study the author investigated the P<sup>0</sup> dependence of Λ using the experi-

To explain the experimental Λ values in the P<sup>0</sup> range from the threshold power

where Φ<sup>0</sup> and ζ are constants and Φ<sup>n</sup> is the calculated Φ value shown in Figure 8.

On the other hand, the relationship between the nonlinearity parameter ε and P<sup>0</sup>

Using Eq. (5), Φ<sup>0</sup> ¼ 31:5μs, ζ ¼ 3:6, and the Φ<sup>n</sup> values shown in Figure 8, the Λ

As shown in Figure 9, Λ increases abruptly near the threshold power (Pth) and

where χ is a constant and m is the order of the square root of the power difference P<sup>0</sup> � Pth. ε and χ correspond to the induced polarization and nonlinear susceptibility in nonlinear optics, respectively [62]. In the calculation, the author

values were calculated as a function of P0. The calculated results are shown in

increases with increasing P0. The Λ values at P<sup>0</sup> ¼ 2:0–2:5 W satisfy Eq. (7).

Relationship between the interval Λ and the input power P0. The blue and black solid lines were calculated using Eqs. (7) and (5), respectively. The red open circles are the data reported by Todoroki [23, 26].

Figure 9. The blue solid line in Figure 9 is the curve calculated using

which is the first term on the right of Eq. (5).

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>Φ</sup>nð Þ� <sup>ε</sup> <sup>Φ</sup>nð Þ <sup>ε</sup> <sup>¼</sup> <sup>0</sup> <sup>p</sup> Φ<sup>0</sup>

<sup>ε</sup> <sup>¼</sup> <sup>χ</sup>ð Þ <sup>P</sup><sup>0</sup> � Pth ð Þ <sup>m</sup>=<sup>2</sup> , (6)

Λð Þ¼ P<sup>0</sup> Φ0Vfð Þ P<sup>0</sup> , (7)

, (5)

" #

<sup>Φ</sup>nð Þ� <sup>ε</sup> <sup>Φ</sup>nð Þ <sup>ε</sup> <sup>¼</sup> <sup>0</sup> <sup>p</sup> Vfð Þ <sup>P</sup><sup>0</sup> on the right of Eq. (5)

λ<sup>0</sup> ¼ 1:48μm [13, 23].

can be expressed as

adopted χ ¼ 1 and m ¼ 2.

Figure 9.

48

compression of the cladding layer. The increment δr in the radius r of the solid-state cladding layer can be expressed in terms of the Young's modulus E and Poisson's ratio ν of the (solid-state) silica glass, and is given by the following equation [63].

� � ð Þþ <sup>1</sup> � <sup>ν</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>ν</sup>

We consider the tensile stress σθ acting on the inner wall (r ¼ ri) of the cladding

<sup>f</sup> <sup>þ</sup> <sup>r</sup><sup>2</sup> i

σθ increases with increasing p. Using ri � 10μm and rf ¼ 62:5μm, σθ was esti-

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 [64]:

By using Eq. (11) and E ¼ 73 GPa for silica glass, we can estimate σ<sup>0</sup> to be approximately 7.3 GPa. Since this value is larger than the estimated σθ value (2.1 GPa), the cladding layer is never broken, but it can be seen that a relatively large expansion of the inner radius occurs as a result of the internal pressure.

Todoroki reported that dmelted and the diameter d of periodic cavities with Λ � 22μm, which is equal to that in the case of ε ¼ 9 and γ ¼ 0, were about 20 and 6.5 μm, respectively [13]. We adopted ri ¼ dmelted=2 ffi 10μm and rf ¼ 62:5μm. Using E ¼ 73 GPa and Poisson's ratio ν ¼ 0:17 for silica glass, the relationship between <sup>δ</sup>r=ri and <sup>r</sup>=ri at <sup>p</sup> <sup>¼</sup> 2 GPa (=1:<sup>97</sup> � <sup>10</sup><sup>4</sup> atm) is calculated. The results are shown in Figure 13. It can be clearly seen from Figure 13 that the elongation rate

δr=ri of the inner radius has a maximum value (about 3.35%) when r=ri.

σθ <sup>¼</sup> <sup>r</sup><sup>2</sup>

mated to be about 2.1 GPa when p ¼ 2 GPa. If this σθ value exceeds the ideal fracture strength σ<sup>0</sup> of the silica glass, a crack will be generated on the inner wall of

r2 <sup>f</sup> � r<sup>2</sup> i

" #

r2 f r2

� r (9)

� p: (10)

σ0≈E=10: (11)

<sup>δ</sup><sup>r</sup> <sup>¼</sup> <sup>r</sup><sup>2</sup>

DOI: http://dx.doi.org/10.5772/intechopen.81154

E r<sup>2</sup> <sup>f</sup> � <sup>r</sup><sup>2</sup> i

layer. σθ is related to p by the following expression [63]:

the cladding layer.

Figure 13.

51

Relationship between δr=ri and r=ri.

i p

Cavity Generation Modeling of Fiber Fuse in Single-Mode Optical Fibers

Figure 11. Time dependence of the perturbed density during fiber fuse propagation. <sup>ε</sup> <sup>¼</sup> <sup>9</sup>, <sup>β</sup> <sup>¼</sup> <sup>6</sup>:5, <sup>γ</sup> <sup>¼</sup> ‐2.

Figure 12. Relationship between l=Λ and the nonlinearity parameter γ. ε ¼ 9, β ¼ 6:5.

γ ¼ 2. On the other hand, as shown in Figure 11, τ<sup>r</sup> is smaller than τ<sup>c</sup> and l=Λ< 0:5 in the case of <sup>γ</sup> <sup>¼</sup> ‐2.

Figure 12 shows the relationship between l=Λ and the nonlinearity parameter γ. As shown in Figure 12, l=Λ increases with increasing γ and approaches its maximum value (about 0.71) at γ � 2.8. In contrast, l=Λ approaches its minimum value (about 0.29) at γ � �2:8.

### 3.1 Deformation of cladding due to plasma formation

The inside of the high-temperature core of 4,000–10,000 K has a high internal pressure <sup>p</sup> of 1 � 104 –<sup>5</sup> � 104 atm [18]. The inner wall of the core (in the solid state) will be expanded by this internal pressure p. To simplify the calculation, the existence of molten silica glass (liquid state) between the solid-state cladding layer (inner radius ri, outer radius rf ) and the inner high-pressure gas plasma is ignored [33].

ri for the cladding is assumed to be dmelted/2. With increasing inner pressure p, the inner radius of the cladding layer increases in the radial direction owing to the Cavity Generation Modeling of Fiber Fuse in Single-Mode Optical Fibers DOI: http://dx.doi.org/10.5772/intechopen.81154

compression of the cladding layer. The increment δr in the radius r of the solid-state cladding layer can be expressed in terms of the Young's modulus E and Poisson's ratio ν of the (solid-state) silica glass, and is given by the following equation [63].

$$\delta r = \frac{r\_i^2 p}{E \left(r\_f^2 - r\_i^2\right)} \left[ (\mathbf{1} - \nu) + (\mathbf{1} + \nu) \frac{r\_f^2}{r^2} \right] \cdot r \tag{9}$$

Todoroki reported that dmelted and the diameter d of periodic cavities with Λ � 22μm, which is equal to that in the case of ε ¼ 9 and γ ¼ 0, were about 20 and 6.5 μm, respectively [13]. We adopted ri ¼ dmelted=2 ffi 10μm and rf ¼ 62:5μm. Using E ¼ 73 GPa and Poisson's ratio ν ¼ 0:17 for silica glass, the relationship between <sup>δ</sup>r=ri and <sup>r</sup>=ri at <sup>p</sup> <sup>¼</sup> 2 GPa (=1:<sup>97</sup> � <sup>10</sup><sup>4</sup> atm) is calculated. The results are shown in Figure 13. It can be clearly seen from Figure 13 that the elongation rate δr=ri of the inner radius has a maximum value (about 3.35%) when r=ri.

We consider the tensile stress σθ acting on the inner wall (r ¼ ri) of the cladding layer. σθ is related to p by the following expression [63]:

$$
\sigma\_{\theta} = \frac{r\_f^2 + r\_i^2}{r\_f^2 - r\_i^2} \cdot p. \tag{10}
$$

σθ increases with increasing p. Using ri � 10μm and rf ¼ 62:5μm, σθ was estimated to be about 2.1 GPa when p ¼ 2 GPa. If this σθ value exceeds the ideal fracture strength σ<sup>0</sup> of the silica glass, a crack will be generated on the inner wall of the 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 [64]:

$$
\sigma\_0 \approx \mathbf{E}/\mathbf{10}.\tag{11}
$$

By using Eq. (11) and E ¼ 73 GPa for silica glass, we can estimate σ<sup>0</sup> to be approximately 7.3 GPa. Since this value is larger than the estimated σθ value (2.1 GPa), the cladding layer is never broken, but it can be seen that a relatively large expansion of the inner radius occurs as a result of the internal pressure.

Figure 13. Relationship between δr=ri and r=ri.

γ ¼ 2. On the other hand, as shown in Figure 11, τ<sup>r</sup> is smaller than τ<sup>c</sup> and l=Λ< 0:5 in

Time dependence of the perturbed density during fiber fuse propagation. <sup>ε</sup> <sup>¼</sup> <sup>9</sup>, <sup>β</sup> <sup>¼</sup> <sup>6</sup>:5, <sup>γ</sup> <sup>¼</sup> ‐2.

Fiber Optics - From Fundamentals to Industrial Applications

Figure 12 shows the relationship between l=Λ and the nonlinearity parameter γ. As shown in Figure 12, l=Λ increases with increasing γ and approaches its maximum value (about 0.71) at γ � 2.8. In contrast, l=Λ approaches its minimum value

The inside of the high-temperature core of 4,000–10,000 K has a high internal

will be expanded by this internal pressure p. To simplify the calculation, the existence of molten silica glass (liquid state) between the solid-state cladding layer (inner radius ri, outer radius rf ) and the inner high-pressure gas plasma is ignored [33]. ri for the cladding is assumed to be dmelted/2. With increasing inner pressure p, the inner radius of the cladding layer increases in the radial direction owing to the

–<sup>5</sup> � 104 atm [18]. The inner wall of the core (in the solid state)

the case of <sup>γ</sup> <sup>¼</sup> ‐2.

Figure 12.

Figure 11.

(about 0.29) at γ � �2:8.

pressure <sup>p</sup> of 1 � 104

50

3.1 Deformation of cladding due to plasma formation

Relationship between l=Λ and the nonlinearity parameter γ. ε ¼ 9, β ¼ 6:5.

The excess volume ΔV produced by the expansion of the inner radius over the interval Λ of the cavity can be estimated as follows using the maximum δr value δrmax at r ¼ ri:

$$
\Delta V = \Lambda \pi \left[ \left( r\_i + \delta r\_{\text{max}} \right)^2 - r\_i^2 \right]. \tag{12}
$$

As the maximum elongation rate δrmax=ri was about 3.35% (see Figure 13), δrmax was estimated to be about 0.335 μm by using ri ffi 10μm.

On the other hand, the volume V of a cavity with diameter d and length l is given by

$$V = l\pi \left(\frac{d}{2}\right)^2. \tag{13}$$

As shown in Eq. (15), the allowable value of l=Λ increases with decreasing cavity diameter d. Figure 15 shows the relationship between the maximum allowable value of l=Λ and the diameter d. As shown in Figure 15, when d is reduced by 20% from

Under this condition, cavity pattern (d) (long filaments) in addition to periodic pattern (c) in Figure 2 can be formed in the core. As the number of repetitions of pattern (d) can change freely, the period of long filaments can be irregular. This may be the cause of the long non-periodic filaments observed by several researchers

Kashyap reported that the diameter of a short asymmetric cavity with l=Λ< 0:5 was larger than that of an oblong and cylindrically symmetric cavity with l=Λ of about 0.5 and that the diameter of a long bullet-shaped cavity with l=Λ< 0:5 was smaller than that of the cavities described above [7]. These findings are consistent with the calculation results shown in Figure 15. In what follows, the production and

When gaseous SiO and/or SiO2 molecules are heated to high temperatures of above 5,000 K, they decompose to form Si and O atoms, and finally become Si<sup>þ</sup> and

In a confined core zone, and thus at high pressures, SiO2 is decomposed with the

The number densities NSiO, NSi, and N<sup>O</sup> (in cm�3) can be estimated using the procedure described in [57, 66] and the published thermochemical data [67] for Si,

The dependence of N<sup>O</sup> on the temperature T is shown in Figure 16. N<sup>O</sup> gradually approaches its maximum value (3:<sup>3</sup> � 1021cm�3) at 11,100 K and then decreases

SiO2⇄SiO þ ð Þ 1=2 O2⇄Si þ 2O: (16)

evolution of SiO gas or Si and O atomic gases at elevated temperatures [65]:

diffusion of O2 gas in the high-temperature core layer are described.

3.2 Oxygen production in optical Fiber

O<sup>þ</sup> ions and electrons in the ionized gas plasma state.

l <sup>Λ</sup> <sup>≤</sup>1:

Relationship between the maximum allowable value of l=Λ and the cavity diameter d. dmelted ¼ 20 μm.

Cavity Generation Modeling of Fiber Fuse in Single-Mode Optical Fibers

DOI: http://dx.doi.org/10.5772/intechopen.81154

6.5 to 5.2 μm, we obtain.

[20–22, 32].

Figure 15.

SiO, O, O2, and SiO2.

53

It is considered that the volume required to generate a cavity was compensated by the excess volume ΔV [33]. If the value of V required to generate a cavity in the interval Λ is smaller than ΔV, the oscillation pattern predicted by Eq. (1) will be maintained and periodic cavities having a size corresponding to V will be formed in the core. That is, the necessary condition for the formation of a periodic cavity pattern is that the ratio of V to ΔV is smaller than 1, which is expressed as follows:

$$\frac{V}{\Delta V} = \frac{l}{\Lambda} \frac{d^2}{4\delta r\_{\text{max}} (2r\_i + \delta r\_{\text{max}})} \le 1. \tag{14}$$

Rearranging Eq. (14), we obtain the following inequality for l=Λ:

$$\frac{d}{d\Lambda} \le \frac{4}{d^2} \delta r\_{\text{max}} (2r\_i + \delta r\_{\text{max}}).\tag{15}$$

When ri � 10μm, δrmax � 0:335μm, and d � 6:5μm, we obtain

$$\frac{l}{\Lambda} \le 0.645.$$

When l=Λ satisfies this condition, the periodic cavities predicted by Eq. (1) will be formed in the core.

However, as shown in Figure 12, l=Λ can be larger than 0.645 when γ> 1.5. In this case, the cavities formed in the core will be compressed and deformed as shown in Figure 14.

Figure 14. Schematic view of cavity compression and deformation in core.

Cavity Generation Modeling of Fiber Fuse in Single-Mode Optical Fibers DOI: http://dx.doi.org/10.5772/intechopen.81154

Figure 15. Relationship between the maximum allowable value of l=Λ and the cavity diameter d. dmelted ¼ 20 μm.

As shown in Eq. (15), the allowable value of l=Λ increases with decreasing cavity diameter d. Figure 15 shows the relationship between the maximum allowable value of l=Λ and the diameter d. As shown in Figure 15, when d is reduced by 20% from 6.5 to 5.2 μm, we obtain.

$$\frac{l}{\Lambda} \le 1.$$

Under this condition, cavity pattern (d) (long filaments) in addition to periodic pattern (c) in Figure 2 can be formed in the core. As the number of repetitions of pattern (d) can change freely, the period of long filaments can be irregular. This may be the cause of the long non-periodic filaments observed by several researchers [20–22, 32].

Kashyap reported that the diameter of a short asymmetric cavity with l=Λ< 0:5 was larger than that of an oblong and cylindrically symmetric cavity with l=Λ of about 0.5 and that the diameter of a long bullet-shaped cavity with l=Λ< 0:5 was smaller than that of the cavities described above [7]. These findings are consistent with the calculation results shown in Figure 15. In what follows, the production and diffusion of O2 gas in the high-temperature core layer are described.

### 3.2 Oxygen production in optical Fiber

When gaseous SiO and/or SiO2 molecules are heated to high temperatures of above 5,000 K, they decompose to form Si and O atoms, and finally become Si<sup>þ</sup> and O<sup>þ</sup> ions and electrons in the ionized gas plasma state.

In a confined core zone, and thus at high pressures, SiO2 is decomposed with the evolution of SiO gas or Si and O atomic gases at elevated temperatures [65]:

$$\text{SiO}\_2 \middle\# \text{SiO} + (\text{1/2})\text{O}\_2 \middle\# \text{Si} + 2\text{O}.\tag{16}$$

The number densities NSiO, NSi, and N<sup>O</sup> (in cm�3) can be estimated using the procedure described in [57, 66] and the published thermochemical data [67] for Si, SiO, O, O2, and SiO2.

The dependence of N<sup>O</sup> on the temperature T is shown in Figure 16. N<sup>O</sup> gradually approaches its maximum value (3:<sup>3</sup> � 1021cm�3) at 11,100 K and then decreases

The excess volume ΔV produced by the expansion of the inner radius over the interval Λ of the cavity can be estimated as follows using the maximum δr value

h i

2 i

: (12)

: (13)

≤1: (14)

<sup>d</sup><sup>2</sup> <sup>δ</sup>rmaxð Þ <sup>2</sup>ri <sup>þ</sup> <sup>δ</sup>rmax : (15)

<sup>Δ</sup><sup>V</sup> <sup>¼</sup> <sup>Λ</sup><sup>π</sup> ð Þ ri <sup>þ</sup> <sup>δ</sup>rmax <sup>2</sup> � <sup>r</sup>

On the other hand, the volume V of a cavity with diameter d and length l is

<sup>V</sup> <sup>¼</sup> <sup>l</sup><sup>π</sup> <sup>d</sup>

was estimated to be about 0.335 μm by using ri ffi 10μm.

Fiber Optics - From Fundamentals to Industrial Applications

V <sup>Δ</sup><sup>V</sup> <sup>¼</sup> <sup>l</sup> Λ

> l Λ ≤ 4

As the maximum elongation rate δrmax=ri was about 3.35% (see Figure 13), δrmax

2 � �<sup>2</sup>

It is considered that the volume required to generate a cavity was compensated by the excess volume ΔV [33]. If the value of V required to generate a cavity in the interval Λ is smaller than ΔV, the oscillation pattern predicted by Eq. (1) will be maintained and periodic cavities having a size corresponding to V will be formed in the core. That is, the necessary condition for the formation of a periodic cavity pattern is that the ratio of V to ΔV is smaller than 1, which is expressed as follows:

> d2 4δrmaxð Þ 2ri þ δrmax

Rearranging Eq. (14), we obtain the following inequality for l=Λ:

When ri � 10μm, δrmax � 0:335μm, and d � 6:5μm, we obtain

l

<sup>Λ</sup> <sup>≤</sup> <sup>0</sup>:645:

When l=Λ satisfies this condition, the periodic cavities predicted by Eq. (1) will

However, as shown in Figure 12, l=Λ can be larger than 0.645 when γ> 1.5. In this case, the cavities formed in the core will be compressed and deformed as shown

δrmax at r ¼ ri:

given by

be formed in the core.

in Figure 14.

Figure 14.

52

Schematic view of cavity compression and deformation in core.

Figure 16. Temperature dependences of the number densities of O and Oþ.

with further increasing T. This is because oxygen (O) atoms are ionized to produce O<sup>þ</sup> ions and electrons in the ionized gas plasma as follows:

$$\mathbf{O} \not\Rightarrow \mathbf{O}^+ + \mathbf{e}^-. \tag{17}$$

The number density N<sup>O</sup><sup>þ</sup> of O<sup>þ</sup> ions can be estimated using the Saha equation [66, 68]:

$$\frac{N\_{\rm O^{+}}^{2}}{N\_{\rm O}} \approx 2 \frac{(2\pi m\_{\rm e} kT)^{3/2}}{h^{3}} \frac{Z\_{+}}{Z\_{0}} \exp\left(-I\_{p}/k\_{\rm B}T\right),\tag{18}$$

where Ip (= 13.61 eV [69]) is the ionization energy of a neutral O atom, me is the electron mass, h is Planck's constant, and kB is Boltzmann's constant. Z<sup>þ</sup> and Z<sup>0</sup> are the partition functions of ionized atoms and neutral atoms, respectively, and Zþ≈Z0. The relationship between N<sup>O</sup><sup>þ</sup> and T is also shown in Figure 16. N<sup>O</sup><sup>þ</sup> increases gradually at temperatures above 7,000 K and reaches 8:<sup>9</sup> � <sup>10</sup><sup>21</sup> cm�<sup>3</sup> at <sup>2</sup> � <sup>10</sup><sup>4</sup> K.

It has been found that molecular oxygen is released and remains in the cavities of a damaged core layer while maintaining a relatively high pressure (about 4 atm [7] or 5–10 atm [20]) at room temperature. The molecular oxygen (O2) is produced from neutral O atoms as follows:

$$\mathbf{2O} \to \mathbf{O}\_2.\tag{19}$$

Figure 18 shows the temperature distribution of the high-temperature front along the z direction at t ¼ 3 ms after the incidence of 1.8 W laser light for IA ¼ 8 dB. The calculation of the temperature distribution was described in Ref. [72]. In Figure 18 the initial attenuation IA of 8 dB corresponds to an optical absorption coefficient <sup>α</sup> of 1:<sup>84</sup> � <sup>10</sup><sup>6</sup> <sup>m</sup>�<sup>1</sup> when the thickness of the absorption layer, which consists of carbon black, is about 1 μm [72]. In this figure, the center of the hightemperature front is set at L ¼ 0 μm. As shown in Figure 18, ΔLs, which is about 36.5 μm, is the distance between the high-temperature peak (L ¼ 0 μm) and the

Temperature distribution of the high-temperature front versus the length along the z direction at t ¼ 3 ms after the incidence of 1.8 W laser light for IA ¼ 8 dB. The center of the high-temperature front is set at L ¼ 0 μm.

This ΔLs can be converted into the time lag Δτ<sup>s</sup> from the passage of the high-

<sup>Δ</sup>τ<sup>s</sup> <sup>¼</sup> <sup>Δ</sup>Ls Vf

: (21)

location with a temperature of 12,700 K.

temperature front as follows:

Figure 17.

Figure 18.

55

Temperature dependence of the production rate of O2.

DOI: http://dx.doi.org/10.5772/intechopen.81154

Cavity Generation Modeling of Fiber Fuse in Single-Mode Optical Fibers

The rate equation of this reaction is [70]

$$\frac{dN\_{\rm O\_2}}{dt} = \sqrt{2}\pi\sigma^2 \sqrt{\frac{8RT}{\pi\mathcal{M}\_\bullet}} N\_\bullet^{-2} \exp\left(-E\_a/RT\right),\tag{20}$$

where <sup>σ</sup> (= 1.5 Å) is half of the collision diameter, <sup>M</sup><sup>O</sup> (<sup>¼</sup> <sup>16</sup>:<sup>0</sup> � <sup>10</sup>�<sup>3</sup> kg) is the atomic weight of O, and Ea is the activation energy. The bond energy (493.6 kJ/mol [71]) of oxygen was used for Ea.

The dependence of dNO2=dt on the temperature T is shown in Figure 17. The rate of O2 production dNO2=dt exhibits its maximum value (2:<sup>96</sup> � <sup>10</sup><sup>31</sup> cm�3s�1) at 12,700 K. This means that the oxygen molecules are produced most effectively at 12,700 K.

Cavity Generation Modeling of Fiber Fuse in Single-Mode Optical Fibers DOI: http://dx.doi.org/10.5772/intechopen.81154

Figure 17. Temperature dependence of the production rate of O2.

Figure 18.

with further increasing T. This is because oxygen (O) atoms are ionized to produce

The number density N<sup>O</sup><sup>þ</sup> of O<sup>þ</sup> ions can be estimated using the Saha equation

Zþ Z0

where Ip (= 13.61 eV [69]) is the ionization energy of a neutral O atom, me is the electron mass, h is Planck's constant, and kB is Boltzmann's constant. Z<sup>þ</sup> and Z<sup>0</sup> are the partition functions of ionized atoms and neutral atoms, respectively, and Zþ≈Z0. The relationship between N<sup>O</sup><sup>þ</sup> and T is also shown in Figure 16. N<sup>O</sup><sup>þ</sup> increases gradually at temperatures above 7,000 K and reaches 8:<sup>9</sup> � <sup>10</sup><sup>21</sup> cm�<sup>3</sup> at

It has been found that molecular oxygen is released and remains in the cavities of a damaged core layer while maintaining a relatively high pressure (about 4 atm [7] or 5–10 atm [20]) at room temperature. The molecular oxygen (O2) is produced

> ffiffiffiffiffiffiffiffiffiffi 8RT πM<sup>O</sup>

N<sup>O</sup>

where <sup>σ</sup> (= 1.5 Å) is half of the collision diameter, <sup>M</sup><sup>O</sup> (<sup>¼</sup> <sup>16</sup>:<sup>0</sup> � <sup>10</sup>�<sup>3</sup> kg) is the atomic weight of O, and Ea is the activation energy. The bond energy (493.6 kJ/mol

The dependence of dNO2=dt on the temperature T is shown in Figure 17. The rate of O2 production dNO2=dt exhibits its maximum value (2:<sup>96</sup> � <sup>10</sup><sup>31</sup> cm�3s�1) at 12,700 K. This means that the oxygen molecules are produced most effectively at

r

ð Þ <sup>2</sup>πmekT <sup>3</sup>=<sup>2</sup> h3

O⇄O<sup>þ</sup> þ e�: (17)

2O ! O2: (19)

<sup>2</sup> exp ð Þ �Ea=RT , (20)

exp �Ip=kBT � �, (18)

O<sup>þ</sup> ions and electrons in the ionized gas plasma as follows:

N2 Oþ N<sup>O</sup> ≈2

Temperature dependences of the number densities of O and Oþ.

Fiber Optics - From Fundamentals to Industrial Applications

The rate equation of this reaction is [70]

dNO2 dt <sup>¼</sup> ffiffi 2 <sup>p</sup> πσ<sup>2</sup>

[66, 68]:

Figure 16.

<sup>2</sup> � <sup>10</sup><sup>4</sup> K.

12,700 K.

54

from neutral O atoms as follows:

[71]) of oxygen was used for Ea.

Temperature distribution of the high-temperature front versus the length along the z direction at t ¼ 3 ms after the incidence of 1.8 W laser light for IA ¼ 8 dB. The center of the high-temperature front is set at L ¼ 0 μm.

Figure 18 shows the temperature distribution of the high-temperature front along the z direction at t ¼ 3 ms after the incidence of 1.8 W laser light for IA ¼ 8 dB. The calculation of the temperature distribution was described in Ref. [72]. In Figure 18 the initial attenuation IA of 8 dB corresponds to an optical absorption coefficient <sup>α</sup> of 1:<sup>84</sup> � <sup>10</sup><sup>6</sup> <sup>m</sup>�<sup>1</sup> when the thickness of the absorption layer, which consists of carbon black, is about 1 μm [72]. In this figure, the center of the hightemperature front is set at L ¼ 0 μm. As shown in Figure 18, ΔLs, which is about 36.5 μm, is the distance between the high-temperature peak (L ¼ 0 μm) and the location with a temperature of 12,700 K.

This ΔLs can be converted into the time lag Δτ<sup>s</sup> from the passage of the hightemperature front as follows:

$$
\Delta \mathbf{r}\_s = \frac{\Delta L\_s}{V\_f}.\tag{21}
$$

Figure 19. Δτ<sup>s</sup> values versus t after the incidence of 1.8 W laser light for IA ¼ 8 dB.

It is expected that the O2 molecular gas in the ionized gas plasma will be observed most frequently after a time lag of Δτ<sup>s</sup> from the passage of the hightemperature peak. If the produced O2 gas diffuses into the rarefied part of the oscillatory variation in density shown in Figures 4–6, 10, and 11, periodic cavities containing some of the oxygen molecules will be formed (see below).

When Vf ¼ 1 m/s, the Δτ<sup>s</sup> values were estimated at a time of t ¼ 1:55–3 ms after the incidence of 1.8 W laser light for IA ¼ 8 dB. The calculated Δτ<sup>s</sup> values are plotted in Figure 19 as a function of t. The fiber fuse phenomenon was initiated at t ¼ 1:5 ms (see Figure 14 in Ref. [72]). As shown in Figure 19, Δτ<sup>s</sup> increases rapidly with increasing t immediately after the fiber fuse is initiated and reaches a constant value (36.5 μs) at t> 1.65 ms. This value is in reasonable agreement with the experimental values (20–70 μs) reported by Dianov and coworkers [30, 31].

### 3.3 Diffusion length of oxygen gas

The O2 gas produced near the high-temperature front diffuses from the compressed part into the rarefied part of the oscillatory variation during a short period Φ of 10–30 μs (see Figure 8).

The diffusion coefficient D of the O2 gas is given by [70].

$$D = \frac{2}{3\pi\sigma^2 N\_{\text{O}\_2}} \sqrt{\frac{RT}{\pi M\_{\text{O}\_2}}},\tag{22}$$

This Δz value is of the same order as the observed periodic cavity interval

Schematic view of diffusion of oxygen gas from the compressed part into the rarefied part in the high-

Cavity Generation Modeling of Fiber Fuse in Single-Mode Optical Fibers

DOI: http://dx.doi.org/10.5772/intechopen.81154

Figure 20 shows a schematic view of the diffusion of the O2 gas from the compressed part into the rarefied part in the high-temperature plasma. If the absolute value of Δz is larger than half of the interval Λ between the periodic rarefied parts, many of the O2 molecules produced in the compressed part can move into the rarefied part during the period Φ (10–30 μs) of the relaxation oscillation. This O2 gas will form temporary microscopic cavities that can constitute the nuclei neces-

As described above, the nonlinear oscillation model was able to phenomenolog-

The evolution of a fiber fuse in a single-mode optical fiber was studied theoretically. To clarify both the silica-glass densification and cavity formation, which are observed in fiber fuse propagation, we investigated a nonlinear oscillation model using the Van der Pol equation. This model was able to phenomenologically explain the densification of the core material, the formation of periodic cavities, the cavity shape, and the regularity of the cavity pattern in the core layer as a result of the

This nonlinear oscillation model including the relaxation oscillation is a phenomenological model, and the relationship between the nonlinearity parameters (ε, β, γ) and the physical properties observed in the fiber fuse experiments is unknown. Therefore, to clarify this relationship, further quantitative investigation

A. Electrostatic interaction between charged surface and plasma

In a confined core zone, and thus at a high pressure, SiO2 is decomposed with the evolution of SiO gas or Si and O atomic gases at elevated temperatures, as described in the main text. When the Si and O atomic gases are heated to high

ically explain both the densification of the core material and the formation of periodic cavities in the core layer as a result of the relaxation oscillation and the

(13–22 μm) [13].

temperature plasma.

Figure 20.

4. Conclusion

is necessary.

57

sary for growth into macroscopic bubbles [74].

formation of O2 gas near the high-temperature front.

relaxation oscillation and cavity compression and/or deformation.

where <sup>M</sup>O2 (<sup>¼</sup> <sup>32</sup>:<sup>0</sup> � <sup>10</sup>�<sup>3</sup> kg) is the molecular weight of O2 gas. As <sup>N</sup>O2 is smaller than NO=2, NO2≈NO/2 is assumed in the calculation.

The mean square of the displacement Δz<sup>2</sup> along the z direction of the optical fiber can be estimated from D and time t as follows [73]:

$$
\Delta \overline{z}^2 = 2Dt.\tag{23}
$$

The Δz values at T ¼ 12, 700 K were estimated using Eqs. (16) and (17). When t ¼ 20μs, the calculated Δz value is given by

$$
\Delta \overline{z} = \pm 16.7 \,\mu m.
$$

Cavity Generation Modeling of Fiber Fuse in Single-Mode Optical Fibers DOI: http://dx.doi.org/10.5772/intechopen.81154

#### Figure 20.

It is expected that the O2 molecular gas in the ionized gas plasma will be observed most frequently after a time lag of Δτ<sup>s</sup> from the passage of the hightemperature peak. If the produced O2 gas diffuses into the rarefied part of the oscillatory variation in density shown in Figures 4–6, 10, and 11, periodic cavities

the incidence of 1.8 W laser light for IA ¼ 8 dB. The calculated Δτ<sup>s</sup> values are plotted in Figure 19 as a function of t. The fiber fuse phenomenon was initiated at t ¼ 1:5 ms (see Figure 14 in Ref. [72]). As shown in Figure 19, Δτ<sup>s</sup> increases rapidly with increasing t immediately after the fiber fuse is initiated and reaches a constant

value (36.5 μs) at t> 1.65 ms. This value is in reasonable agreement with the experimental values (20–70 μs) reported by Dianov and coworkers [30, 31].

The diffusion coefficient D of the O2 gas is given by [70].

smaller than NO=2, NO2≈NO/2 is assumed in the calculation.

fiber can be estimated from D and time t as follows [73]:

t ¼ 20μs, the calculated Δz value is given by

<sup>D</sup> <sup>¼</sup> <sup>2</sup> 3πσ<sup>2</sup>NO2

3.3 Diffusion length of oxygen gas

Φ of 10–30 μs (see Figure 8).

56

Figure 19.

When Vf ¼ 1 m/s, the Δτ<sup>s</sup> values were estimated at a time of t ¼ 1:55–3 ms after

The O2 gas produced near the high-temperature front diffuses from the compressed part into the rarefied part of the oscillatory variation during a short period

where <sup>M</sup>O2 (<sup>¼</sup> <sup>32</sup>:<sup>0</sup> � <sup>10</sup>�<sup>3</sup> kg) is the molecular weight of O2 gas. As <sup>N</sup>O2 is

The mean square of the displacement Δz<sup>2</sup> along the z direction of the optical

The Δz values at T ¼ 12, 700 K were estimated using Eqs. (16) and (17). When

Δz ¼ �16:7 μm:

ffiffiffiffiffiffiffiffiffiffiffi RT πMO2

, (22)

<sup>Δ</sup>z<sup>2</sup> <sup>¼</sup> <sup>2</sup>Dt: (23)

s

containing some of the oxygen molecules will be formed (see below).

Δτ<sup>s</sup> values versus t after the incidence of 1.8 W laser light for IA ¼ 8 dB.

Fiber Optics - From Fundamentals to Industrial Applications

Schematic view of diffusion of oxygen gas from the compressed part into the rarefied part in the hightemperature plasma.

This Δz value is of the same order as the observed periodic cavity interval (13–22 μm) [13].

Figure 20 shows a schematic view of the diffusion of the O2 gas from the compressed part into the rarefied part in the high-temperature plasma. If the absolute value of Δz is larger than half of the interval Λ between the periodic rarefied parts, many of the O2 molecules produced in the compressed part can move into the rarefied part during the period Φ (10–30 μs) of the relaxation oscillation. This O2 gas will form temporary microscopic cavities that can constitute the nuclei necessary for growth into macroscopic bubbles [74].

As described above, the nonlinear oscillation model was able to phenomenologically explain both the densification of the core material and the formation of periodic cavities in the core layer as a result of the relaxation oscillation and the formation of O2 gas near the high-temperature front.
