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

temperatures of above 3,000 K (Si) and 4,000 K (O), they are ionized to produce Si<sup>þ</sup> and O<sup>þ</sup> ions and electrons in the ionized gas plasma state.

$$\text{Si} + \text{O} \middle\sharp \text{Si}^+ + \text{O}^+ + 2\text{e}^- \tag{24}$$

If thermally produced electrons in the plasma are not bound to positive species (Si<sup>þ</sup> or O<sup>þ</sup> ions), they can move freely in the plasma under the action of the alternating electric field of the light wave. Such free diffusion is possible only in the limiting case of very low charge densities. However, as shown in Figure 16 and also Figure 1 in Ref. [66], the densities of Si<sup>þ</sup> and O<sup>þ</sup> ions and electrons are reasonably large above 1 � 104 K. At high charge densities, it is known that the positive and negative species diffuse at the same rate. This phenomenon, proposed by Schottky [75], is called ambipolar diffusion [76, 77]. Ambipolar diffusion is the diffusion of positive and negative species owing to their interaction via an electric field (spacecharge field). In plasma physics, ambipolar diffusion is closely related to the concept of quasineutrality.

Some electrons arrive at the surface of melted silica glass, and they attach to oxygen atoms on the surface because oxygen atoms have a high electron affinity [78]. As a result, a negatively charged surface, which was proposed by Yakovlenko [33], may be formed as shown in Figure 21.

However, the negative charges on the surface will immediately be balanced by an equal number of oppositely charged Si<sup>þ</sup> and O<sup>þ</sup> ions because these positive ions move together with the electrons as a result of ambipolar diffusion. In this way, an atmosphere of ions is formed in the rapid thermal motion close to the surface. This ionic atmosphere is known as the diffuse electric double layer [79].

The thickness δ<sup>0</sup> of the double layer is approximately 1/κ, which is the characteristic length known as the Debye length. The parameter κ is given in terms of N<sup>e</sup> and T as follows [77]:

$$
\kappa^2 = \frac{2N\_\text{e}e^2}{\varepsilon\_0 k\_B T},
\tag{25}
$$

where e is the charge of an electron and ε<sup>0</sup> is the dielectric constant of vacuum.

When <sup>T</sup> <sup>¼</sup> <sup>1</sup> � <sup>10</sup><sup>4</sup> K, <sup>N</sup><sup>e</sup> <sup>¼</sup> <sup>2</sup>:<sup>2</sup> � <sup>10</sup><sup>20</sup> cm�3. Using these values and Eq. (25), the thickness <sup>δ</sup><sup>0</sup> of the double layer at 1 � <sup>10</sup><sup>4</sup> K was estimated to be about

Schematic view of the cross section of the high-temperature plasma in the optical fiber.

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

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

double layers is schematically shown in Figure 22.

B. Nonlinearity parameter β in Van der pol equation

propagation can be represented by the Van der Pol equation

ρ€<sup>1</sup> � ε 1 � βρ<sup>1</sup>

A cross section of the high-temperature plasma in the optical fiber with the

In the central domain of the high-temperature plasma, electrically neutral atoms (Si and O) and charged species (Siþ, Oþ, and e�) exist. As the charged species are balanced, electrical neutrality is achieved in the domain. Moreover, the dimensions of the domain are almost equal to those of the high-temperature plasma excluding the very thin (Å order) electric double layers at the surface of the melted silica glass.

The dynamical behavior of the perturbed density ρ<sup>1</sup> resulting from fiber fuse

2

, ρ\_ <sup>1</sup> ¼ dρ1=dt, ε and β are nonlinearity parameters, and the

ρ<sup>1</sup> ¼ A cosð Þ ω0t þ φ , (27)

ρ<sup>1</sup> ¼ 0, (26)

: (28)

<sup>2</sup> <sup>ρ</sup>\_ <sup>1</sup> <sup>þ</sup> <sup>ω</sup><sup>0</sup>

where the amplitude A and phase φ are slowly varying functions, then A satisfies

<sup>2</sup> <sup>þ</sup> <sup>ρ</sup>\_ <sup>1</sup> ω0 <sup>2</sup>

<sup>A</sup><sup>2</sup> <sup>¼</sup> <sup>ρ</sup><sup>1</sup>

<sup>3</sup>:<sup>3</sup> � <sup>10</sup>�<sup>10</sup> m.

Figure 22.

where <sup>ρ</sup>€<sup>1</sup> <sup>¼</sup> <sup>d</sup><sup>2</sup>

the following equation:

59

ρ1=dt<sup>2</sup>

nonlinearity parameter γ ¼ 0 is assumed. If the solution of Eq. (26) is written as

Differentiating Eq. (28), we obtain

Figure 21. Schematic view of the negatively charged surface and ionic atmosphere.

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

#### Figure 22.

temperatures of above 3,000 K (Si) and 4,000 K (O), they are ionized to produce

If thermally produced electrons in the plasma are not bound to positive species

Some electrons arrive at the surface of melted silica glass, and they attach to oxygen atoms on the surface because oxygen atoms have a high electron affinity [78]. As a result, a negatively charged surface, which was proposed by Yakovlenko

However, the negative charges on the surface will immediately be balanced by an equal number of oppositely charged Si<sup>þ</sup> and O<sup>þ</sup> ions because these positive ions move together with the electrons as a result of ambipolar diffusion. In this way, an atmosphere of ions is formed in the rapid thermal motion close to the surface. This

The thickness δ<sup>0</sup> of the double layer is approximately 1/κ, which is the characteristic length known as the Debye length. The parameter κ is given in terms of N<sup>e</sup>

<sup>κ</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>Nee<sup>2</sup>

<sup>ε</sup>0kBT , (25)

ionic atmosphere is known as the diffuse electric double layer [79].

Schematic view of the negatively charged surface and ionic atmosphere.

(Si<sup>þ</sup> or O<sup>þ</sup> ions), they can move freely in the plasma under the action of the alternating electric field of the light wave. Such free diffusion is possible only in the limiting case of very low charge densities. However, as shown in Figure 16 and also Figure 1 in Ref. [66], the densities of Si<sup>þ</sup> and O<sup>þ</sup> ions and electrons are reasonably large above 1 � 104 K. At high charge densities, it is known that the positive and negative species diffuse at the same rate. This phenomenon, proposed by Schottky [75], is called ambipolar diffusion [76, 77]. Ambipolar diffusion is the diffusion of positive and negative species owing to their interaction via an electric field (spacecharge field). In plasma physics, ambipolar diffusion is closely related to the con-

Si þ O⇄Si<sup>þ</sup> þ O<sup>þ</sup> þ 2e� (24)

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

Fiber Optics - From Fundamentals to Industrial Applications

cept of quasineutrality.

and T as follows [77]:

Figure 21.

58

[33], may be formed as shown in Figure 21.

Schematic view of the cross section of the high-temperature plasma in the optical fiber.

where e is the charge of an electron and ε<sup>0</sup> is the dielectric constant of vacuum. When <sup>T</sup> <sup>¼</sup> <sup>1</sup> � <sup>10</sup><sup>4</sup> K, <sup>N</sup><sup>e</sup> <sup>¼</sup> <sup>2</sup>:<sup>2</sup> � <sup>10</sup><sup>20</sup> cm�3. Using these values and Eq. (25), the thickness <sup>δ</sup><sup>0</sup> of the double layer at 1 � <sup>10</sup><sup>4</sup> K was estimated to be about <sup>3</sup>:<sup>3</sup> � <sup>10</sup>�<sup>10</sup> m.

A cross section of the high-temperature plasma in the optical fiber with the double layers is schematically shown in Figure 22.

In the central domain of the high-temperature plasma, electrically neutral atoms (Si and O) and charged species (Siþ, Oþ, and e�) exist. As the charged species are balanced, electrical neutrality is achieved in the domain. Moreover, the dimensions of the domain are almost equal to those of the high-temperature plasma excluding the very thin (Å order) electric double layers at the surface of the melted silica glass.
