Open access peer-reviewed chapter

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

By Yoshito Shuto

Submitted: July 10th 2018Reviewed: August 27th 2018Published: November 5th 2018

DOI: 10.5772/intechopen.81154

Downloaded: 275

Abstract

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 relaxation oscillation and cavity compression and/or deformation. Furthermore, the production and diffusion of O2 gas in the high-temperature core layer were described on the basis of the nonlinear oscillation model.

Keywords

  • fiber fuse
  • nonlinear oscillation
  • Van der Pol equation

1. Introduction

Owing to the progress of dense wavelength-division multiplexing (DWDM) technology using an optical-fiber amplifier, we can exchange large amounts of data at a rate of over 100 Tbit/s over several hundred kilometers [1]. However, it is widely recognized that the maximum transmission capacity of a single strand of fiber is rapidly approaching its limit of 100 Tbit/s owing to the optical power limitations imposed by the fiber fuse phenomenon and the finite transmission bandwidth determined by optical-fiber amplifiers [2]. To overcome these limitations, space-division multiplexing (SDM) technology using a multicore fiber (MCF) was proposed [3, 4], and 1 Pbit/s transmission was demonstrated using a low-crosstalk 12-core fiber [5].

The fiber fuse phenomenon was first observed in 1987 by British scientists [6, 7, 8, 9]. Several review articles [10, 11, 12, 13, 14] have been recently published that cover many aspects of the current understanding of fiber fuses.

A fiber fuse can be generated by bringing the end of a fiber into contact with an absorbent material or melting a small region of a fiber using an arc discharge of a fusion splice machine [6, 15, 16, 17]. If a fiber fuse is generated, an intense blue-white flash occurs in the fiber core, and this flash propagates along the core in the direction of the optical power source at a velocity on the order of 1 m/s. The temperature and pressure in the region where this flash occurs have been estimated to be about 104 K and 104 atm, respectively [18]. Fuses are terminated by gradually reducing the laser power to a termination threshold at which the energy balance in the fuse is broken.

The critical diameter dmelted, which is usually larger than the core diameter 2rc, is a characteristic dimensional parameter of the fiber fuse effect. In the inner area with diameter ddmelted, a fiber fuse (high-temperature ionized gas plasma) propagates and silica glass is melted [18]. dmelted, defined as the diameter of the melting area, is considered as the radial size of the plasma generated in the fiber fuse [19]. Dianov et al. reported that the refractive index of the inner area with ddmeltedin Ge-doped and/or pure silica core fibers is increased by silica-glass densification and/or the redistribution of the dopant (Ge) [20].

When a fiber fuse is generated, the core layer in which the fuse propagates is seriously damaged, and the damage has the form of periodic bullet-shaped cavities or non-periodic filaments remaining in the core [6, 7, 8, 9, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] (see Figure 1). Needless to say, the density in a cavity or filament is lower than that of the neighboring silica glass. It has been found that molecular oxygen is released and remains in the cavities while maintaining a high pressure (about 4 atm [7] or 5–10 atm [20]) at room temperature. Recently, several types of sensors based on periodic cavities have been proposed as a cost-effective approach to sensor production [27, 28, 29].

Figure 1.

Schematic view of damaged optical fiber.

The dynamics of cavity formation have been investigated since the discovery of the fiber fuse phenomenon. Dianov and coworkers observed the formation of periodic bullet-shaped cavities 20–70 μs after the passage of a plasma leading edge [30, 31].

Kashyap reported that the cavity shape was dependent on the nature of the input laser light (CW or pulses) operated at a wavelength λ0of 1.064 μm when the average input power was maintained at 2 W [7, 15]. When CW light was input, the cavities appeared to be elliptical and cylindrically symmetric. On the other hand, short asymmetric cavities were formed by injecting (mode-locked) pulses with 100 ps FWHM (full width at half maximum), while long bullet-shaped cavities were observed by injecting pulses with 190 ps FWHM [7, 15]. Hand and Russell reported the appearance of highly regular periodic damage tracks in germanosilicate fibers at λ0=488and 514 nm [9]. Davis et al. reported that long non-periodic filaments occurred in germanium-doped depressed clad fibers, and a periodic damage pattern was observed in fibers doped with phosphorus and germanium at λ0=1.064μm[21, 22]. Atkins et al. observed both periodic and long non-periodic damage tracks created in a germanosilicate-core single-mode fiber transmitting about 2 W of power at 488 nm [32]. Dianov and coworkers reported the formation of periodic damage in a silica-core fiber at 1.064 and 1.21 μm [18, 30, 31] and long non-periodic damage in a germanosilicate silica core fiber at 488 and 514 nm [20].

Todoroki classified fiber fuse propagation into three modes (unstable, unimodal, and cylindrical) according to the plasma volume relative to the pump beam size [26]. When the pump power was increased or decreased rapidly, an increase in the length of the void-free segment or the occurrence of an irregular void pattern was observed, respectively [26].

From these observation results, the cavity patterns occurring in single-mode fibers can be classified into the four patterns shown in Figure 2, where lis the length of the cavity and Λis the (periodic) cavity interval. The observed periodic cavity patterns belong to patterns (a)–(c) with the pattern depending on the value of l/Λ. The long non-periodic cavity pattern (filaments) can be considered as a sequence of two or more of pattern (d).

Figure 2.

Cavity patterns observed in optical fiber.

These cavities have been considered to be the result of either the classic Rayleigh instability caused by the capillary effect in the molten silica surrounding a vaporized fiber core [32] or the electrostatic repulsion between negatively charged layers induced at the plasma–molten silica interface [33, 34]. Although the capillary effect convincingly explains the formation mechanism of water droplets from a tap and/or bubbles through a water flow, this effect does not appear to apply to the cavity formation mechanism of a fiber fuse owing to the anomalously high viscosity of the silica glass [23, 33]. Yakovlenko proposed a novel cavity formation mechanism based on the formation of an electric charge layer on the interface between the liquid glass and plasma [33]. This charge layer, where the electrons adhere to the liquid glass surface, gives rise to a “negative” surface tension coefficient for the liquid layer. In the case of a negative surface tension coefficient, the deformation of the liquid surface proceeds, giving rise to a long bubble that is pressed into the liquid [33]. Furthermore, an increase in the charged surface due to the repulsion of similar charges results in the development of instability [33]. The instability emerges because the countercurrent flowing in the liquid causes the liquid to enter the region filled with plasma, and the extruded liquid forms a bridge. Inside the region separated from the front part of the fuse by this bridge, gas condensation and cooling of the molten silica glass occur [34]. A row of cavities is formed by the repetition of this process. Although Yakovlenko’s explanation of the formation of a long cavity and rows of cavities is very interesting, the concept of “negative” surface tension appears to be unfeasible in the field of surface science and/or plasma physics (see Appendix A).

Low-frequency plasma instabilities are triggered by moving the high-temperature front of a fiber fuse toward the light source. It is well known that such a low-frequency plasma instability behaves as a Van der Pol oscillator with instability frequency ω0[35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. Therefore, the oscillatory motion of the ionized gas plasma during fiber fuse propagation can be studied phenomenologically using the Van der Pol equation [56].

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 parameters in the nonlinear oscillation model is reported.

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 ρof the plasma is assumed to be in the form ρ=ρ0+ρ1, where ρ0is the initial density of the stationary (unperturbed) part in the front region of the plasma and ρ1is the perturbed density. The dynamical behavior of ρ1resulting from fiber fuse propagation can be represented by the Van der Pol equation

d2ρ1dt2ε1βρ12+2γρ1dρ1dt+ω02ρ1=0,E1

where εis a parameter that characterizes the degree of nonlinearity and βcharacterizes the nonlinear saturation (see Appendix B). The nonlinearity parameter γcharacterizes the oscillation pattern.

The angular frequency ω0of the oscillation of the gas plasma is determined by the ion-sound velocity Csand the free-running distance Lfof the ion-sound wave, and is given by

ω0=2πf=2πCsLf.E2

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

Cs=RTeMi,E3

where Ris the gas constant, Teis the temperature of the electron, and Miis the mass of the ion. The author estimated Cs=1300m/sby using Te=5760K, which was the average temperature of the radiation zone [57], and Mi=28×103kgfor a Si+ion. The free-running distance Lfwas 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 ω0values 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=0.1, β=6.5, and γ=0was calculated using Eq. (1). The calculated result is shown in Figure 3, where the perturbed density ρ1is plotted as a function of time. When t80μs, the maximum and minimum values of ρ1for the ionized gas plasma reach 0.86 and − 0.86, respectively. The maximum 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].

Figure 3.

Time dependence of the perturbed density during fiber fuse propagation. ε = 0.1 , β = 6.5 , γ = 0 .

On the other hand, it can be seen that for ε=0.1the motion of the Van der Pol oscillator is very nearly harmonic, exhibiting alternate compression and rarefaction of the density with a relatively small period Φof about 6.3μs.

Next, the oscillatory motion for ε=5, 9, and 14 with β=6.5and γ=0was examined. The calculated results are shown in Figures 4, 5, 6, respectively. It can be seen that for ε=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 Φvalues of the motion corresponding to ε=5, 9, and 14 were estimated to be about 12.9, 21.6, and 36.1 μs, respectively. These Φvalues are much larger than that (about 6.3 μs) for ε=0.1.

Figure 4.

Time dependence of the perturbed density during fiber fuse propagation. ε = 5 , β = 6.5 , γ = 0 .

Figure 5.

Time dependence of the perturbed density during fiber fuse propagation. ε = 9 , β = 6.5 , γ = 0 .

Figure 6.

Time dependence of the perturbed density during fiber fuse propagation. ε = 14 , β = 6.5 , γ = 0 .

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 Vfwhen 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.

Figure 7.

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, Λis the interval between the periodic compressed (or rarefied) parts.

The relationship between the period Φand the interval Λis

Λ=ΦVf,E4

where Vfis the propagation velocity of the fiber fuse and Vf=1m/s was assumed in the calculation. The Λvalues of the motion corresponding to ε=5, 9, and 14 are thus estimated to be about 12.9, 21.6, and 36.1 μm, respectively, using Eq. (4) and Vf=1m/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, Λis equal to the periodic cavity interval. The estimated Λvalues (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 Φand 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 Φand/or Λoccurs 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.

Figure 8.

Relationship between the period Φ and the nonlinearity parameter ε . β = 6.5 , γ = 0 .

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 Λand the input laser power P0observed in fiber fuse propagation are described.

2.1 Power dependence of periodic cavity interval

It is well known that the fiber-fuse propagation velocity Vfincreases with increasing input laser power P0[7, 8, 22, 23, 25, 26, 58, 59, 60]. Furthermore, in addition to Vf, Todoroki reported the P0dependence of Λin an SMF-28e fiber at λ0=1.48μm[13, 23].

In this study the author investigated the P0dependence of Λusing the experimental Vfvalues [23, 26] and the calculated Φvalues shown in Figure 8.

To explain the experimental Λvalues in the P0range from the threshold power (Pth1.3W [61]) to 9 W, ΛP0can be represented by

ΛP0=Φ0VfP01ζΦnεΦnε=0Φ0,E5

where Φ0and ζare constants and Φnis the calculated Φvalue shown in Figure 8.

The second term ζΦnεΦnε=0VfP0on the right of Eq. (5) represents the contribution of the nonlinearity to the overall Λvalue.

On the other hand, the relationship between the nonlinearity parameter εand P0can be expressed as

ε=χP0Pthm/2,E6

where χis a constant and mis the order of the square root of the power difference P0Pth. εand χcorrespond to the induced polarization and nonlinear susceptibility in nonlinear optics, respectively [62]. In the calculation, the author adopted χ=1and m=2.

Using Eq. (5), Φ0=31.5μs, ζ=3.6, and the Φnvalues shown in Figure 8, the Λvalues 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

ΛP0=Φ0VfP0,E7
which is the first term on the right of Eq. (5).

Figure 9.

Relationship between the interval Λ and the input power P 0 . 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].

As shown in Figure 9, Λincreases abruptly near the threshold power (Pth) and increases with increasing P0. The Λvalues at P0=2.02.5W satisfy Eq. (7). However, with increasing P0, the Λvalues at P0>2.5W are less than those calculated using Eq. (7) and approach the Λvalues 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>3.5W [23] or P0>2.3W [26]. As shown in Figure 9, the distinct contribution of the nonlinearity to the overall Λvalue begins at P0of 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.

3. Effect of nonlinearity parameters on cavity patterns

The nonlinearity parameter γcharacterizes the oscillation pattern. The oscillatory motion for ε=9, β=6.5, and γ=0was shown in Figure 5, where the perturbed density ρ1is 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 τrof the rarefied regions equals that of the compressed regions τc. The relationship between the period Φ(=τr+τc) and the interval Λis given by Eq. (4), and the relationship between τrand the length lof the cavity is

l=τrVf.E8

The Λand lvalues of the motion corresponding to ε=9, β=6.5, and γ=0are estimated to be about 10.8 and 21.6 μm, respectively, using Eqs. (4) and (8) and Vf=1m/s. That is, l/Λ=0.5in the case of γ=0.

Next, the oscillatory motion for γ=2and − 2 with ε=9and β=6.5was examined. The calculated results are shown in Figures 10 and 11, respectively. As shown in Figure 10, the retention time τrof the rarefied regions is larger than that of the compressed regions τc. As a result, the ratio l/Λis larger than 0.5 in the case of γ=2. On the other hand, as shown in Figure 11, τris smaller than τcand l/Λ<0.5in the case of γ=2.

Figure 10.

Time dependence of the perturbed density during fiber fuse propagation. ε = 9 , β = 6.5 , γ = 2 .

Figure 11.

Time dependence of the perturbed density during fiber fuse propagation. ε = 9 , β = 6.5 , γ = ‐ 2 .

Figure 12 shows the relationship between l/Λand the nonlinearity parameter γ. As shown in Figure 12, l/Λincreases with increasing γand approaches its maximum value (about 0.71) at γ2.8. In contrast, l/Λapproaches its minimum value (about 0.29) at γ2.8.

Figure 12.

Relationship between l / Λ and the nonlinearity parameter γ . ε = 9 , β = 6.5 .

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 pof 1 × 104–5 × 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].

rifor 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 compression of the cladding layer. The increment δrin the radius rof the solid-state cladding layer can be expressed in terms of the Young’s modulus Eand Poisson’s ratio νof the (solid-state) silica glass, and is given by the following equation [63].

δr=ri2pErf2ri21ν+1+νrf2r2rE9

Todoroki reported that dmeltedand the diameter dof periodic cavities with Λ22μm, which is equal to that in the case of ε=9and γ=0, were about 20 and 6.5 μm, respectively [13]. We adopted ri=dmelted/210μmand rf=62.5μm. Using E=73GPa and Poisson’s ratio ν=0.17for silica glass, the relationship between δr/riand r/riat p=2GPa (=1.97×104atm) is calculated. The results are shown in Figure 13. It can be clearly seen from Figure 13 that the elongation rate δr/riof the inner radius has a maximum value (about 3.35%) when r/ri.

Figure 13.

Relationship between δr / r i and r / r i .

We consider the tensile stress σθacting on the inner wall (r=ri) of the cladding layer. σθis related to pby the following expression [63]:

σθ=rf2+ri2rf2ri2p.E10

σθincreases with increasing p. Using ri10μmand rf=62.5μm, σθwas estimated to be about 2.1 GPa when p=2GPa. If this σθvalue exceeds the ideal fracture strength σ0of 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 σ0value is related to the Young’s modulus Eof the material by the following equation [64]:

σ0E/10.E11

By using Eq. (11) and E=73GPa for silica glass, we can estimate σ0to be approximately 7.3 GPa. Since this value is larger than the estimated σθvalue (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.

The excess volume ΔVproduced by the expansion of the inner radius over the interval Λof the cavity can be estimated as follows using the maximum δrvalue δrmaxat r=ri:

ΔV=Λπri+δrmax2ri2.E12

As the maximum elongation rate δrmax/riwas about 3.35% (see Figure 13), δrmaxwas estimated to be about 0.335 μm by using ri10μm.

On the other hand, the volume Vof a cavity with diameter dand length lis given by

V=d22.E13

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

VΔV=lΛd24δrmax2ri+δrmax1.E14

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

lΛ4d2δrmax2ri+δrmax.E15

When ri10μm, δrmax0.335μm, and d6.5μm, we obtain

lΛ0.645.

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

However, as shown in Figure 12, l/Λcan 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.

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

lΛ1.

Figure 15.

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

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, 21, 22, 32].

Kashyap reported that the diameter of a short asymmetric cavity with l/Λ<0.5was larger than that of an oblong and cylindrically symmetric cavity with l/Λof about 0.5 and that the diameter of a long bullet-shaped cavity with l/Λ<0.5was 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 O2gas in the high-temperature core layer are described.

3.2 Oxygen production in optical Fiber

When gaseous SiO and/or SiO2molecules are heated to high temperatures of above 5,000 K, they decompose to form Si and O atoms, and finally become Si+and O+ions and electrons in the ionized gas plasma state.

In a confined core zone, and thus at high pressures, SiO2is decomposed with the evolution of SiO gas or Si and O atomic gases at elevated temperatures [65]:

SiO2SiO+1/2O2Si+2O.E16

The number densities NSiO, NSi, and NO(in cm3) 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 NOon the temperature Tis shown in Figure 16. NOgradually approaches its maximum value (3.3×1021cm3) at 11,100 K and then decreases with further increasing T. This is because oxygen (O) atoms are ionized to produce O+ions and electrons in the ionized gas plasma as follows:

OO++e.E17

Figure 16.

Temperature dependences of the number densities of O and O + .

The number density NO+of O+ions can be estimated using the Saha equation [66, 68]:

NO+2NO22πmekT3/2h3Z+Z0expIp/kBT,E18

where Ip(= 13.61 eV [69]) is the ionization energy of a neutral O atom, meis the electron mass, his Planck’s constant, and kBis Boltzmann’s constant. Z+and Z0are the partition functions of ionized atoms and neutral atoms, respectively, and Z+Z0. The relationship between NO+and Tis also shown in Figure 16. NO+increases gradually at temperatures above 7,000 K and reaches 8.9×1021cm3at 2×104K.

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:

2OO2.E19

The rate equation of this reaction is [70]

dNO2dt=2πσ28RTπMONO2expEa/RT,E20

where σ(= 1.5 Å) is half of the collision diameter, MO(=16.0×103kg) is the atomic weight of O, and Eais the activation energy. The bond energy (493.6 kJ/mol [71]) of oxygen was used for Ea.

The dependence of dNO2/dton the temperature Tis shown in Figure 17. The rate of O2production dNO2/dtexhibits its maximum value (2.96×1031cm3s1) at 12,700 K. This means that the oxygen molecules are produced most effectively at 12,700 K.

Figure 17.

Temperature dependence of the production rate of O 2 .

Figure 18 shows the temperature distribution of the high-temperature front along the zdirection at t=3ms after the incidence of 1.8 W laser light for IA=8dB. The calculation of the temperature distribution was described in Ref. [72]. In Figure 18 the initial attenuation IAof 8 dB corresponds to an optical absorption coefficient αof 1.84×106m1when the thickness of the absorption layer, which consists of carbon black, is about 1 μm [72]. In this figure, the center of the high-temperature front is set at L=0μm. As shown in Figure 18, ΔLs, which is about 36.5 μm, is the distance between the high-temperature peak (L=0μm) and the location with a temperature of 12,700 K.

Figure 18.

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

This ΔLscan be converted into the time lag Δτsfrom the passage of the high-temperature front as follows:

Δτs=ΔLsVf.E21

It is expected that the O2molecular gas in the ionized gas plasma will be observed most frequently after a time lag of Δτsfrom the passage of the high-temperature peak. If the produced O2gas diffuses into the rarefied part of the oscillatory variation in density shown in Figures 4, 5, 6, 10, and 11, periodic cavities containing some of the oxygen molecules will be formed (see below).

When Vf=1m/s, the Δτsvalues were estimated at a time of t=1.553ms after the incidence of 1.8 W laser light for IA=8dB. The calculated Δτsvalues are plotted in Figure 19 as a function of t. The fiber fuse phenomenon was initiated at t=1.5ms (see Figure 14 in Ref. [72]). As shown in Figure 19, Δτsincreases rapidly with increasing timmediately 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].

Figure 19.

Δ τ s values versus t after the incidence of 1.8 W laser light for IA = 8 dB.

3.3 Diffusion length of oxygen gas

The O2gas produced near the high-temperature front diffuses from the compressed part into the rarefied part of the oscillatory variation during a short period Φof 10–30 μs (see Figure 8).

The diffusion coefficient Dof the O2gas is given by [70].

D=23πσ2NO2RTπMO2,E22

where MO2(=32.0×103kg) is the molecular weight of O2gas. As NO2is smaller than NO/2, NO2NO/2 is assumed in the calculation.

The mean square of the displacement Δz¯2along the zdirection of the optical fiber can be estimated from Dand time tas follows [73]:

Δz¯2=2Dt.E23

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

Δz¯=±16.7 μm.

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 O2gas 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 Λbetween the periodic rarefied parts, many of the O2molecules produced in the compressed part can move into the rarefied part during the period Φ(10–30 μs) of the relaxation oscillation. This O2gas will form temporary microscopic cavities that can constitute the nuclei necessary for growth into macroscopic bubbles [74].

Figure 20.

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

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 O2gas near the high-temperature front.

4. Conclusion

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 relaxation oscillation and cavity compression and/or deformation.

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 is necessary.

In a confined core zone, and thus at a high pressure, SiO2is 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+and O+ions and electrons in the ionized gas plasma state.

Si+OSi++O++2eE24

If thermally produced electrons in the plasma are not bound to positive species (Si+or O+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+and O+ions and electrons are reasonably large above 1×104K. 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 (space-charge 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.

Figure 21.

Schematic view of the negatively charged surface and ionic atmosphere.

However, the negative charges on the surface will immediately be balanced by an equal number of oppositely charged Si+and O+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 δ0of the double layer is approximately 1/κ, which is the characteristic length known as the Debye length. The parameter κis given in terms of Neand Tas follows [77]:

κ2=2Nee2ε0kBT,E25

where eis the charge of an electron and ε0is the dielectric constant of vacuum. When T=1×104K, Ne=2.2×1020cm3. Using these values and Eq. (25), the thickness δ0of the double layer at 1×104K was estimated to be about 3.3×1010m.

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

Figure 22.

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

In the central domain of the high-temperature plasma, electrically neutral atoms (Si and O) and charged species (Si+, O+, 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 ρ1resulting from fiber fuse propagation can be represented by the Van der Pol equation

ρ¨1ε1βρ12ρ̇1+ω02ρ1=0,E26

where ρ¨1=d2ρ1/dt2, ρ̇1=dρ1/dt, εand βare nonlinearity parameters, and the nonlinearity parameter γ=0is assumed.

If the solution of Eq. (26) is written as

ρ1=Acosω0t+φ,E27

where the amplitude Aand phase φare slowly varying functions, then Asatisfies the following equation:

A2=ρ12+ρ̇1ω02.E28

Differentiating Eq. (28), we obtain

Ȧ=ρ̇1ω02Aρ¨1+ω02ρ1=ρ̇1ω02Aε1βρ12ρ̇1=εω02Aρ̇12εβω02Aρ12ρ̇12=εAsin2ω0t+φεβA3sin2ω0t+φcos2ω0t+φ=ε2A1cos2ω0t+2φεβ8A31cos4ω0t+4φ.E29

Because of the slowly varying property of A, the oscillatory terms Acos2ω0t+2φand A3cos4ω0t+4φon the right of Eq. (29) are averaged out every cycle and can be discarded [80], thus reducing Eq. (29) to

Ȧε2Aεβ8A3ε2A1β4A2.E30

The maximum value of A, Am, is obtained under the condition of Ȧ=0. To satisfy this condition,

Am=2β.E31

This means that the nonlinearity parameter βdetermines the maximum and minimum values of ρ1. In the calculation, we used β=6.5, which corresponds to Am0.8.

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Yoshito Shuto (November 5th 2018). Cavity Generation Modeling of Fiber Fuse in Single-Mode Optical Fibers, Fiber Optics - From Fundamentals to Industrial Applications, Patrick Steglich and Fabio De Matteis, IntechOpen, DOI: 10.5772/intechopen.81154. Available from:

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