2. Nonlinear oscillation behavior in ionized gas plasma

An ionized gas plasma exhibits oscillatory motion with a small amplitude when the high-temperature front of a fiber fuse propagates toward the light source.

The density ρ of the plasma is assumed to be in the form ρ ¼ ρ<sup>0</sup> þ ρ1, where ρ<sup>0</sup> is the initial density of the stationary (unperturbed) part in the front region of the plasma and ρ<sup>1</sup> is the perturbed density. The dynamical behavior of ρ<sup>1</sup> resulting from fiber fuse propagation can be represented by the Van der Pol equation

$$\frac{d^2\rho\_1}{dt^2} - \varepsilon (\mathbf{1} - \beta \rho\_1 \mathbf{1}^2 + 2\eta \rho\_1) \frac{d\rho\_1}{dt} + a\rho\_0 \,^2\rho\_1 = \mathbf{0},\tag{1}$$

where ε is a parameter that characterizes the degree of nonlinearity and β characterizes the nonlinear saturation (see Appendix B). The nonlinearity parameter γ characterizes the oscillation pattern.

The angular frequency ω<sup>0</sup> of the oscillation of the gas plasma is determined by the ion-sound velocity Cs and the free-running distance Lf of the ion-sound wave, and is given by

$$
\rho\_0 = 2\pi f = 2\pi \frac{C\_s}{L\_f}.\tag{2}
$$

where f is the frequency of the oscillation of the gas plasma. The ion-sound velocity Cs is given by [38]

$$\mathbf{C}\_{s} = \sqrt{\frac{RT\_{e}}{\mathbf{M}\_{i}}},\tag{3}$$

value (0.86) means that the increase in density of the core material reaches 86%, which is almost equal to the experimental value (87%) estimated by Dianov et al. [20]. On the other hand, it can be seen that for ε ¼ 0:1 the motion of the Van der Pol oscillator is very nearly harmonic, exhibiting alternate compression and rarefaction

Time dependence of the perturbed density during fiber fuse propagation. ε ¼ 5, β ¼ 6:5, γ ¼ 0.

Time dependence of the perturbed density during fiber fuse propagation. ε ¼ 0:1, β ¼ 6:5, γ ¼ 0.

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

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

Next, the oscillatory motion for ε ¼ 5, 9, and 14 with β ¼ 6:5 and γ ¼ 0 was examined. The calculated results are shown in Figures 4–6, respectively. It can be seen that for ε ¼ 5, 9, and 14, the oscillations consist of sudden transitions between compressed and rarefied regions. This type of motion is called a relaxation oscillation [56]. The Φ values of the motion corresponding to ε ¼ 5, 9, and 14 were estimated to be about 12.9, 21.6, and 36.1 μs, respectively. These Φ values are much

The oscillatory motion generated in the high-temperature front of the ionized gas plasma can be transmitted to the neighboring plasma at the rate of Vf when the fiber fuse propagates toward the light source. Figure 7 shows a schematic view of the dimensional relationship between the temperature and the perturbed density of

In Figure 7, Λ is the interval between the periodic compressed (or rarefied)

of the density with a relatively small period Φ of about 6:3μs.

larger than that (about 6.3 μs) for ε ¼ 0:1.

parts.

45

Figure 3.

Figure 4.

the ionized gas plasma during fiber fuse propagation.

where R is the gas constant, Te is the temperature of the electron, and Mi is the mass of the ion. The author estimated Cs ¼ 1300 m=s by using Te ¼ 5760 K, which was the average temperature of the radiation zone [57], and Mi <sup>¼</sup> <sup>28</sup> � <sup>10</sup>�<sup>3</sup> kg for a Si<sup>þ</sup> ion. The free-running distance Lf was assumed to be 1.3 mm, which was almost equal to the distance (about 1.5 mm [57]) of the radiation zone. Using Eq. (2) and the Cs (= 1300 m/s) and Lf (= 1.3 mm) values, the frequency f of the oscillation was estimated to be about 1 MHz. The relatively high f or ω<sup>0</sup> values reported in the literature were 426–620 kHz [52, 53] and 14.5–40.9 MHz [35, 42, 45]. These relatively high frequencies are owing to the excitation of high-frequency electron oscillation together with ion oscillation in the ionized gas plasma. The f value (= 1 MHz) estimated above is comparable to these experimental values.

The oscillatory motion for ε ¼ 0:1, β ¼ 6:5, and γ ¼ 0 was calculated using Eq. (1). The calculated result is shown in Figure 3, where the perturbed density ρ<sup>1</sup> is plotted as a function of time. When t≥80μs, the maximum and minimum values of ρ<sup>1</sup> for the ionized gas plasma reach 0.86 and � 0.86, respectively. The maximum

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

Figure 3. Time dependence of the perturbed density during fiber fuse propagation. ε ¼ 0:1, β ¼ 6:5, γ ¼ 0.

Figure 4. Time dependence of the perturbed density during fiber fuse propagation. ε ¼ 5, β ¼ 6:5, γ ¼ 0.

value (0.86) means that the increase in density of the core material reaches 86%, which is almost equal to the experimental value (87%) estimated by Dianov et al. [20].

On the other hand, it can be seen that for ε ¼ 0:1 the motion of the Van der Pol oscillator is very nearly harmonic, exhibiting alternate compression and rarefaction of the density with a relatively small period Φ of about 6:3μs.

Next, the oscillatory motion for ε ¼ 5, 9, and 14 with β ¼ 6:5 and γ ¼ 0 was examined. The calculated results are shown in Figures 4–6, respectively. It can be seen that for ε ¼ 5, 9, and 14, the oscillations consist of sudden transitions between compressed and rarefied regions. This type of motion is called a relaxation oscillation [56]. The Φ values of the motion corresponding to ε ¼ 5, 9, and 14 were estimated to be about 12.9, 21.6, and 36.1 μs, respectively. These Φ values are much larger than that (about 6.3 μs) for ε ¼ 0:1.

The oscillatory motion generated in the high-temperature front of the ionized gas plasma can be transmitted to the neighboring plasma at the rate of Vf when the fiber fuse propagates toward the light source. Figure 7 shows a schematic view of the dimensional relationship between the temperature and the perturbed density of the ionized gas plasma during fiber fuse propagation.

In Figure 7, Λ is the interval between the periodic compressed (or rarefied) parts.

plasma during fiber fuse propagation can be studied phenomenologically using the

parameters in the nonlinear oscillation model is reported.

Fiber Optics - From Fundamentals to Industrial Applications

dt<sup>2</sup> � <sup>ε</sup> <sup>1</sup> � βρ<sup>1</sup>

d2 ρ1

ter γ characterizes the oscillation pattern.

and is given by

44

velocity Cs is given by [38]

2. Nonlinear oscillation behavior in ionized gas plasma

In this paper the author describes a novel nonlinear oscillation model using the Van der Pol equation and qualitatively explains both the silica-glass densification and cavity formation observed in fiber fuse propagation. Furthermore, an investigation of the relationship between several cavity patterns and the nonlinearity

An ionized gas plasma exhibits oscillatory motion with a small amplitude when

The density ρ of the plasma is assumed to be in the form ρ ¼ ρ<sup>0</sup> þ ρ1, where ρ<sup>0</sup> is the initial density of the stationary (unperturbed) part in the front region of the plasma and ρ<sup>1</sup> is the perturbed density. The dynamical behavior of ρ<sup>1</sup> resulting from

dt <sup>þ</sup> <sup>ω</sup><sup>0</sup>

Lf

2

ρ<sup>1</sup> ¼ 0 , (1)

: (2)

, (3)

the high-temperature front of a fiber fuse propagates toward the light source.

<sup>2</sup> <sup>þ</sup> <sup>2</sup>γρ<sup>1</sup> � � <sup>d</sup>ρ<sup>1</sup>

where ε is a parameter that characterizes the degree of nonlinearity and β characterizes the nonlinear saturation (see Appendix B). The nonlinearity parame-

The angular frequency ω<sup>0</sup> of the oscillation of the gas plasma is determined by the ion-sound velocity Cs and the free-running distance Lf of the ion-sound wave,

<sup>ω</sup><sup>0</sup> <sup>¼</sup> <sup>2</sup>π<sup>f</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> Cs

where f is the frequency of the oscillation of the gas plasma. The ion-sound

r

where R is the gas constant, Te is the temperature of the electron, and Mi is the mass of the ion. The author estimated Cs ¼ 1300 m=s by using Te ¼ 5760 K, which was the average temperature of the radiation zone [57], and Mi <sup>¼</sup> <sup>28</sup> � <sup>10</sup>�<sup>3</sup> kg for a Si<sup>þ</sup> ion. The free-running distance Lf was assumed to be 1.3 mm, which was almost equal to the distance (about 1.5 mm [57]) of the radiation zone. Using Eq. (2) and the Cs (= 1300 m/s) and Lf (= 1.3 mm) values, the frequency f of the oscillation was estimated to be about 1 MHz. The relatively high f or ω<sup>0</sup> values reported in the literature were 426–620 kHz [52, 53] and 14.5–40.9 MHz [35, 42, 45]. These relatively high frequencies are owing to the excitation of high-frequency electron oscillation together with ion oscillation in the ionized gas plasma. The f value (= 1 MHz)

The oscillatory motion for ε ¼ 0:1, β ¼ 6:5, and γ ¼ 0 was calculated using Eq. (1). The calculated result is shown in Figure 3, where the perturbed density ρ<sup>1</sup> is plotted as a function of time. When t≥80μs, the maximum and minimum values of ρ<sup>1</sup> for the ionized gas plasma reach 0.86 and � 0.86, respectively. The maximum

ffiffiffiffiffiffiffiffi RTe Mi

Cs ¼

estimated above is comparable to these experimental values.

fiber fuse propagation can be represented by the Van der Pol equation

Van der Pol equation [56].

The relationship between the period Φ and the interval Λ is

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

μm observed in fiber fuse propagation [13, 23].

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

enhanced with increasing pump power.

2.1 Power dependence of periodic cavity interval

Relationship between the period Φ and the nonlinearity parameter ε. β ¼ 6:5, γ ¼ 0.

Figure 8.

47

where Vf is the propagation velocity of the fiber fuse and Vf ¼ 1 m/s was assumed in the calculation. The Λ values of the motion corresponding to ε ¼ 5, 9, and 14 are thus estimated to be about 12.9, 21.6, and 36.1 μm, respectively, using Eq. (4) and Vf ¼ 1 m/s. If a large amount of molecular oxygen (O2) accumulates in the rarefied part, the periodic formation of bubbles (or cavities) will be observed. In such a case, Λ is equal to the periodic cavity interval. The estimated Λ values (12.9, 21.6, and 36.1 μm) are close to the experimental periodic cavity intervals of 13–22

Figure 8 shows the relationship between Φ and the nonlinearity parameter ε. As

Kashyap reported that the cavity shape was dependent on the nature of the input laser light (CW or pulses) [7, 15]. Todoroki classified the damage to the front part of a fiber fuse into three shapes (two spheroids and a long partially cylindrical cavity) depending on the pump power [23]. He also found that a rapid increase or decrease in the pump power results in an increase in the length of the cavity-free segment or the occurrence of an irregular cavity pattern, respectively [26]. These findings indicate that the cavity shape and the regularity of the cavity pattern may be determined by the degree of nonlinearity of the Van der Pol oscillator.

In what follows, the results of examining the relationship between the interval Λ

and the input laser power P<sup>0</sup> observed in fiber fuse propagation are described.

It is well known that the fiber-fuse propagation velocity Vf increases with increasing input laser power P<sup>0</sup> [7, 8, 22, 23, 25, 26, 58–60]. Furthermore, in

shown in Figure 8, Φ, which is proportional to the interval Λ, increases with increasing ε. That is, the increase in Φ and/or Λ occurs because of the enhanced nonlinearity. It was found that the experimental periodic cavity interval increases

with the laser pump power [13, 23]. It can therefore be presumed that the nonlinearity of the Van der Pol oscillator occurring in the ionized gas plasma is

Λ ¼ ΦVf , (4)

Figure 5. Time dependence of the perturbed density during fiber fuse propagation. ε ¼ 9, β ¼ 6:5, γ ¼ 0.

Figure 6. Time dependence of the perturbed density during fiber fuse propagation. ε ¼ 14, β ¼ 6:5, γ ¼ 0.

#### Figure 7.

Schematic view of the dimensional relationship between the temperature and the perturbed density of the ionized gas plasma during fiber fuse propagation.

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

The relationship between the period Φ and the interval Λ is

$$
\Lambda = \Phi V\_f,\tag{4}
$$

where Vf is the propagation velocity of the fiber fuse and Vf ¼ 1 m/s was assumed in the calculation. The Λ values of the motion corresponding to ε ¼ 5, 9, and 14 are thus estimated to be about 12.9, 21.6, and 36.1 μm, respectively, using Eq. (4) and Vf ¼ 1 m/s. If a large amount of molecular oxygen (O2) accumulates in the rarefied part, the periodic formation of bubbles (or cavities) will be observed. In such a case, Λ is equal to the periodic cavity interval. The estimated Λ values (12.9, 21.6, and 36.1 μm) are close to the experimental periodic cavity intervals of 13–22 μm observed in fiber fuse propagation [13, 23].

Figure 8 shows the relationship between Φ and the nonlinearity parameter ε. As shown in Figure 8, Φ, which is proportional to the interval Λ, increases with increasing ε. That is, the increase in Φ and/or Λ occurs because of the enhanced nonlinearity. It was found that the experimental periodic cavity interval increases with the laser pump power [13, 23]. It can therefore be presumed that the nonlinearity of the Van der Pol oscillator occurring in the ionized gas plasma is enhanced with increasing pump power.

Kashyap reported that the cavity shape was dependent on the nature of the input laser light (CW or pulses) [7, 15]. Todoroki classified the damage to the front part of a fiber fuse into three shapes (two spheroids and a long partially cylindrical cavity) depending on the pump power [23]. He also found that a rapid increase or decrease in the pump power results in an increase in the length of the cavity-free segment or the occurrence of an irregular cavity pattern, respectively [26]. These findings indicate that the cavity shape and the regularity of the cavity pattern may be determined by the degree of nonlinearity of the Van der Pol oscillator.

In what follows, the results of examining the relationship between the interval Λ and the input laser power P<sup>0</sup> observed in fiber fuse propagation are described.

### 2.1 Power dependence of periodic cavity interval

It is well known that the fiber-fuse propagation velocity Vf increases with increasing input laser power P<sup>0</sup> [7, 8, 22, 23, 25, 26, 58–60]. Furthermore, in

Figure 8. Relationship between the period Φ and the nonlinearity parameter ε. β ¼ 6:5, γ ¼ 0.

Figure 6.

Figure 5.

Figure 7.

46

ionized gas plasma during fiber fuse propagation.

Time dependence of the perturbed density during fiber fuse propagation. ε ¼ 14, β ¼ 6:5, γ ¼ 0.

Time dependence of the perturbed density during fiber fuse propagation. ε ¼ 9, β ¼ 6:5, γ ¼ 0.

Fiber Optics - From Fundamentals to Industrial Applications

Schematic view of the dimensional relationship between the temperature and the perturbed density of the

addition to Vf , Todoroki reported the P<sup>0</sup> dependence of Λ in an SMF-28e fiber at λ<sup>0</sup> ¼ 1:48μm [13, 23].

In this study the author investigated the P<sup>0</sup> dependence of Λ using the experimental Vf values [23, 26] and the calculated Φ values shown in Figure 8.

To explain the experimental Λ values in the P<sup>0</sup> range from the threshold power (Pth ≃1:3W [61]) to 9 W, Λð Þ P<sup>0</sup> can be represented by

$$
\Lambda(P\_0) = \Phi\_0 V\_f(P\_0) \left[ \mathbf{1} - \zeta \frac{\sqrt{\Phi\_n(\varepsilon) - \Phi\_n(\varepsilon = 0)}}{\Phi\_0} \right], \tag{5}
$$

where Φ<sup>0</sup> and ζ are constants and Φ<sup>n</sup> is the calculated Φ value shown in Figure 8. The second term �<sup>ζ</sup> 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> Vfð Þ <sup>P</sup><sup>0</sup> on the right of Eq. (5)

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

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

$$
\varepsilon = \chi (P\_0 - P\_{th})^{(m/2)},\tag{6}
$$

However, with increasing P0, the Λ values at P0>2:5 W are less than those calculated

This may be related to the modes of fiber fuse propagation reported by Todoroki [23, 26]. Todoroki classified the damage to the front part of a fiber fuse into three shapes (two spheroids and a long partially cylindrical cavity) depending on the pump power, and the appearance of the long partially cylindrical cavity was observed at P0>3:5 W [23] or P0>2:3 W [26]. As shown in Figure 9, the distinct contribution of the nonlinearity to the overall Λ value begins at P<sup>0</sup> of 2.3–3.5 W, and the oscillatory motion of the gas plasma changes from a nearly harmonic oscillation (see Figure 3) to a relaxation oscillation (see Figure 4) with increasing P0. Therefore, the change from the spheroids of unstable and unimodal modes to the long partially cylindrical cavities of the cylindrical mode may be related to the contribu-

The nonlinearity parameter γ characterizes the oscillation pattern. The oscilla-

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

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

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

l ¼ τrVf : (8)

tory 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

using Eq. (7) and approach the Λ values estimated using Eq. (5).

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

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

3. Effect of nonlinearity parameters on cavity patterns

Vf ¼ 1 m/s. That is, l=Λ ¼ 0:5 in the case of γ ¼ 0.

tion of the nonlinearity.

cavity is

Figure 10.

49

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 adopted χ ¼ 1 and m ¼ 2.

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 values were calculated as a function of P0. The calculated results are shown in Figure 9. The blue solid line in Figure 9 is the curve calculated using

$$
\Lambda(P\_0) = \Phi\_0 V\_f(P\_0),
\tag{7}
$$

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

As shown in Figure 9, Λ increases abruptly near the threshold power (Pth) and increases with increasing P0. The Λ values at P<sup>0</sup> ¼ 2:0–2:5 W satisfy Eq. (7).

Figure 9.

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].

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

However, with increasing P0, the Λ values at P0>2:5 W are less than those calculated using Eq. (7) and approach the Λ values estimated using Eq. (5).

This may be related to the modes of fiber fuse propagation reported by Todoroki [23, 26]. Todoroki classified the damage to the front part of a fiber fuse into three shapes (two spheroids and a long partially cylindrical cavity) depending on the pump power, and the appearance of the long partially cylindrical cavity was observed at P0>3:5 W [23] or P0>2:3 W [26]. As shown in Figure 9, the distinct contribution of the nonlinearity to the overall Λ value begins at P<sup>0</sup> of 2.3–3.5 W, and the oscillatory motion of the gas plasma changes from a nearly harmonic oscillation (see Figure 3) to a relaxation oscillation (see Figure 4) with increasing P0. Therefore, the change from the spheroids of unstable and unimodal modes to the long partially cylindrical cavities of the cylindrical mode may be related to the contribution of the nonlinearity.
