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

The evolution of a fiber-fuse phenomenon in a single-mode optical fiber was studied theoretically. To clarify both the silica-glass densification and cavity formation, which have been observed in fiber fuse propagation, we investigated a nonlinear oscillation model using the Van Der Pol equation. This 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 a relaxation oscillation.


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 60 Tbit/s [1].However, it is widely recognized that the maximum transmission capacity of a single strand of fiber is rapidly approaching its limit of about 100 Tbit/s owing to the optical power limitations imposed by the fiberfuse 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 by using a lowcrosstalk 12-core fiber [5].
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].If a fiber fuse is generated, an intense bluewhite 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 10 4 K and 10 4 atmospheres, respectively [17].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  cr , which is usually larger than the core diameter 2   , is a characteristic dimensional parameter of the fiber fuse effect.In an inner area with diameter  ≤  cr , a fiber fuse (high-temperature plasma) propagates and silica glass is melted [17]. cr , defined as the diameter of the melting area, is considered as the radial size of the plasma generated in the fiber fuse [18].Dianov et al. reported that the refractive index of the inner area with  ≤  cr in Ge-doped and/or pure silica core fibers is increased by silica-glass densification and/or the redistribution of the dopant (Ge) [19].
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 or nonperiodic bullet-shaped cavities remaining in the core [20][21][22][23][24][25][26][27][28] (see Figure 1).Needless to say, the density in a cavity 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 atmospheres [7] or 5-10 atmospheres [19]) at room temperature.Recently, several types of sensors based on periodic cavities have been proposed as a cost-effective approach to sensor production [26][27][28].The dynamics of cavity formation have been investigated since the discovery of the fiber-fuse phenomenon.Kashyap reported that the cavity shape was dependent on the nature of the input laser light (CW or pulses) when the average input power was maintained at 2 W [7].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 bulletshaped cavities were observed by injecting pulses with 190 ps FWHM [7].Dianov and coworkers observed the formation of periodic bullet-shaped cavities 20-70 s after the passage of a plasma leading edge [29,30].Todoroki classified fiberfuse propagation into three modes (unstable, unimodal, and cylindrical) according to the plasma volume relative to the pump beam size [25].When the pump power was increased or decreased rapidly, the increment in length of the void-free segment or the occurrence of an irregular void pattern was observed, respectively [25].
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 [31] or the electrostatic repulsion between negatively charged layers induced at the plasma-molten silica interface [32,33].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 [22,32].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 [32].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 [32].Furthermore, an increase in the charged surface due to the repulsion of similar charges results in the development of instability [32].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 [33].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  0 [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52].
Therefore, the oscillatory motion of the ionized gas plasma during fiber-fuse propagation can be studied phenomenologically using the Van Der Pol equation [53].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.

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 frequency  0 of the oscillation of the gas plasma is determined by the fiber-fuse propagation velocity   and the free-running distance   of the oscillator in the front region of the plasma and is given by The plasma instability is enhanced in the vicinity of the high-temperature peak.Therefore, in this study, the author estimated   from the calculated temperature distribution of the high-temperature front of an ionized gas plasma.Figure 2 shows the temperature distribution of the high-temperature front as a function of the length  along the  direction.The calculation of the temperature distribution was described in [54].In Figure 2, an initial attenuation IA of 8 dB corresponds to an optical absorption coefficient  of 1.84 × 10 6 m −1 when the thickness of the absorption layer, which consists of carbon black, is about 1 m [54].As shown in Figure 2,   was estimated to be about 12.5 m, where   = 1 m/s was used in the calculation.
Here, the density  is considered in the form  =  0 +  1 , where  0 is the initial density of the stationary (unperturbed) part in the front region of the plasma and  1 is the perturbed density.The dynamical behavior of  1 resulting from fiberfuse propagation can be represented by the Van Der Pol equation: where  is a parameter that characterizes the degree of nonlinearity and  characterizes the nonlinear saturation (see Appendix B).
The oscillatory motion for  = 0.1 and  = 6.5 is shown in Figure 3, where the perturbed density  1 is plotted as a function of time.When  ≥ 80 s, the maximum and minimum values of  1 for 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. [19].
On the other hand, it can be seen that for  = 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 6.3 s.
Next, the oscillatory motion for  = 5, 9, and 14 with  = 6.5 was examined.The calculated results are shown in Figures 4-6, respectively.
It can be seen that, for  = 5, 9, and 14, the oscillations consist of sudden transitions between compressed and rarefied regions.This type of motion is called a relaxation oscillation [53], and the term "self-excited oscillation" is also often used.
The Φ values of the motion corresponding to  = 5, 9, and 14 were estimated to be about 12.9, 21.6, and 36.1 s, respectively.These Φ values are much larger than those for  = 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   when the fiber fuse propagates toward the light source.Figure 7 shows 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.
The relationship between the period Φ and the interval Λ is The Λ values of the motion corresponding to  = 5, 9, and 14 are thus estimated to be about 12.9, 21.6, and 36.1 m, respectively, using (3) and   = 1 m/s.If a large amount of molecular oxygen (O 2 ) accumulates in the rarefied part, the periodic formation of bubbles (or voids) will be observed.
In such a case, Λ is equal to the periodic void interval.The estimated Λ values (12.9, 21.6, and 36.1 m) are close to the experimental periodic void intervals of 13-22 m observed in fiber-fuse propagation [22,55].
Figure 8 shows the relationship between Φ and the nonlinearity parameter .As shown in Figure 8, Φ, which is proportional to the interval Λ, increases with increasing .That is, the increase in Φ and/or Λ occurs because of the enhanced nonlinearity.It was found that the experimental periodic void interval increases with the laser pump power [22,55].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].Todoroki classified the damage to the front part of a fiber fuse into three shapes (two spheroids and a long partially cylindrical void) depending on the pump power [22].He also found that fast increment or decrement of the pump power results in the increment in length of the void-free segment or the occurrence of an irregular void pattern, respectively [25].These findings indicate that the void shape and the regularity of the void pattern may be determined by the degree of nonlinearity of the Van Der Pol oscillator.
In what follows, the author describes an examination result about the relationship between the interval Λ and the input laser power  0 observed in fiber-fuse propagation.Figure 8: Relationship between the period Φ and the nonlinearity parameter . = 6.5.
In this study, the author investigated the  0 dependence of Λ using the experimental   values [22,25] and the calculated Φ values shown in Figure 8.
To explain the experimental Λ values in the  0 range from the threshold power ( th ≃ 1.3 W [59]) to 9 W, Λ( 0 ) can be represented by the following equation: where Φ 0 and  are constants and Φ  is the calculated Φ value shown in Figure 8.The second term −√Φ  () − Φ  ( = 0)  ( 0 ) on the right-hand side of (4) represents the contribution of the nonlinearity to the overall Λ value.
On the other hand, the relationship between the nonlinear parameter  and  0 can be expressed as follows: where  is a constant and  is the order of the square root of the power difference  0 −  th . and  correspond to the induced polarization and nonlinear susceptibility in nonlinear optics, respectively [60].In the calculation, the author adopted  = 1 and  = 2.By using (4), Φ 0 = 31.5s,  = 3.6, and the Φ  values shown in Figure 8, the Λ values were calculated as a function of  0 .: Relationship between the interval Λ and the input power  0 .The blue and black solid lines were calculated by using ( 6) and ( 4), respectively.The red open circles are the data reported by Todoroki [22,25].
The calculated results are shown in Figure 9.The blue solid line in Figure 9 was the calculation curve using the following equation: which is the first term on the right-hand side of (4).As shown in Figure 9, Λ exhibits a steep rise near the threshold power ( th ) and increases with increasing  0 .The Λ values at  0 = 2.0-2.5 W obey (6).However, with increasing  0 , the Λ values at  0 of >2.5 W are less than those calculated using (6) and approach the Λ values estimated by using (4).
This may be related to the modes of fiber-fuse propagation reported by Todoroki [22,25].Todoroki classified the damage to the front part of a fiber fuse into three shapes (two spheroids and a long partially cylindrical void) depending on the pump power, and the appearance of the long partially cylindrical void was observed at  0 of >3.5 W [22] or 2.3 W [25].As shown in Figure 9, the distinct contribution of the nonlinearity to the overall Λ value begins at  0 of 2.3-3.5 W, and the oscillatory motion of the gas plasma will change from a nearly harmonic oscillation (see Figure 3) to a relaxation one (see Figure 4) with increasing  0 .Therefore, the change from the spheroids of unstable and unimodal modes to the long partially cylindrical voids of the cylindrical mode may be related to the contribution of the nonlinearity.
As shown in Figures 4-6, rarefaction occurs periodically near the high-temperature peak, and the density of the rarefied gas plasma decreases over a period of Φ/2.
In what follows, the production and diffusion of O 2 gas in the high-temperature core layer are described.

Oxygen Production in Optical
Fiber.When gaseous SiO and/or SiO 2 molecules 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, SiO 2 is decomposed with the evolution of SiO gas or Si and O atomic gases at elevated temperatures [61]: The number densities  SiO ,  Si , and  O (in cm −3 ) can be estimated using the procedure described in [62,63] and the published thermochemical data [64] for Si, SiO, O, O 2 , and SiO 2 .
The dependence of  O on the temperature  is shown in Figure 10. O gradually approaches its maximum value (3.3 × 10 21 cm −3 ) at 11,100 K and then decreases with further increasing .This is because oxygen (O) atoms are ionized to produce O + ions and electrons in the ionized gas plasma as follows: The number density  O + of O + ions can be estimated using the Saha equation [63,65]: where   (= 13.61 eV [66]) is the ionization energy of a neutral O atom,   is the electron mass, ℎ is Planck's constant, and   is Boltzmann's constant. + and  0 are the partition functions of ionized atoms and neutral atoms, respectively, and  + ≈  0 .The relationship between  O + and  is also shown in Figure 10. O + increases gradually at temperatures above 7,000 K and reaches 8.9 × 10 21 cm −3 at 2 × 10 4 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 atmospheres [7] Advances in OptoElectronics or 5-10 atmospheres [19]) at room temperature.The molecular oxygen (O 2 ) is produced from neutral O atoms as follows: The rate equation of this reaction is [67] where  (= 1.5 Å) is half of the collision diameter,  O (= 16.0 × 10 −3 kg) is the atomic weight of O, and   is the activation energy.The bond energy (493.6 kJ/mol [68]) of oxygen was used for   .The dependence of  O 2 / on the temperature  is shown in Figure 11.The rate of O 2 production  O 2 / exhibits its maximum value (2.96 × 10 31 cm −3 s −1 ) at 12,700 K.This means that the oxygen molecules are produced most effectively at 12,700 K.
Figure 12 shows the temperature distribution of the hightemperature front along the  direction at  = 3 ms after the incidence of 1.8 W laser light for IA = 8 dB.In this figure, the center of the high-temperature front is set at  = 0 m.In Figure 12, Δ  , which is about 36.5 m, is the distance between the high-temperature peak ( = 0 m) and the location with a temperature of 12,700 K.
This Δ  can be converted into the time lag Δ  from the passage of the high-temperature front as follows: It is expected that the O 2 molecular gas in the ionized gas plasma will be observed most frequently after a time lag of Δ  from the passage of the high-temperature peak.
If the produced O 2 gas diffuses into the rarefied part of the oscillatory variation in density shown in Figures 4-6, periodic cavities containing some of the oxygen molecules will be formed (see below).When   = 1 m/s, the Δ  values were estimated at a time of  = 1.55-3 ms after the incidence of 1.8 W laser light for IA = 8 dB.The calculated Δ  values are plotted in Figure 13 as a function of .The fiber-fuse phenomenon was initiated at  = 1.5 ms (see Figure 14 in [54]).As shown in Figure 13, Δ  increases rapidly with increasing  immediately after the fiber fuse is initiated and reaches a constant value (36.5 s) at  > 1.65 ms.This value is in reasonable agreement with the experimental values (20-70 s) reported by Dianov and coworkers [29,30].

Diffusion Length of Oxygen Gas.
The O 2 gas produced near the high-temperature front diffuses from the compressed part into the rarefied part of the oscillatory variation during a small period Φ of 10-30 s (see Figure 9).
The diffusion coefficient  of the O 2 gas is given by [67] where The mean square of the displacement Δ 2 along the  direction of the optical fiber can be estimated from  and time  as follows [69]: The Δ values at  = 12,700 K were estimated using ( 13) and (14).When  = 20 s, the calculated Δ value is given by This Δ value is of the same order as the observed periodic void interval (13-22 m) [55].Figure 14 shows a schematic view of the diffusion of the O 2 gas from the compressed part into the rarefied part in the high-temperature plasma.If the absolute value of Δ is larger than half of the interval Λ between the periodic rarefied parts, many of the O 2 molecules produced in the compressed part can move into the rarefied part during the period Φ (10-30 s) of the relaxation oscillation.This O 2 gas will form temporary microscopic voids that can constitute the nuclei necessary for growth into macroscopic bubbles [70].
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 O 2 gas near the high-temperature front.

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 both the densification of the core material and the formation of periodic cavities in the core layer as a result of a relaxation oscillation.
Keen and Fletcher reported that the method of "asynchronous quenching of a Van Der Pol oscillator" might be used to suppress (or quench) a plasma instability [37].This method relies on driving the previously existing Van Der Pol oscillator (or instability) at a high frequency  such that  ≫  0 , where  0 is the frequency of the Van Der Pol oscillator.With increasing the drive amplitude, the system behaves as if the previously existing relaxation oscillation ( 0 ) was destroyed (or quenched) by the asynchronous action of this frequency ().
In the fiber-fuse experiment, the driving frequency might be related to the optical pulse widths of the high-power laser.It will be necessary to clarify the relationship between the driving frequency and the suppression of cavities in detail in the fiber-fuse experiments.
This nonlinear oscillation model including the relaxation oscillation is a phenomenological model, and the relationship between the nonlinearity parameters (, ) and the physical properties observed in the fiber-fuse experiments is unknown.Therefore, to clarify this relationship, further quantitative investigation is also necessary.

A. Electrostatic Interaction between Charged Surface and Plasma
In a confined core zone, and thus at a high pressure, SiO 2 is decomposed with the evolution of SiO gas or Si and O atomic gases at elevated temperatures, as described above.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: 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 10 and also Figure 1 in [63], the densities of Si + and O + ions and electrons are reasonably large above 1 × 10 4 K.At high charge densities, it is known that the positive and negative species diffuse at the same rate.This phenomenon, proposed by Schottky [71], is called ambipolar diffusion [72,73].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 Advances in OptoElectronics because oxygen atoms have a high electron affinity [74].As a result, a negatively charged surface, which was proposed by Yakovlenko [32], may be formed as shown in Figure 15.
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 [75].
The thickness  0 of the double layer is approximately 1/, which is the characteristic length known as the Debye length.The parameter  is given in terms of   and  as follows [73]: where  is the charge of an electron and  0 is the dielectric constant of vacuum.When  = 1 × 10 4 K and   = 2.2 × 10 20 cm −3 .Using these values and (A.2), the thickness  0 of the double layer at 1 × 10 4 K was estimated to be about 3.3 × 10 −10 m.The cross section of the high-temperature plasma in the optical fiber can be schematically illustrated using the double layers as shown in Figure 16.
In the central domain of the high-temperature plasma, electrically neutral atoms (Si and O) and charged species (Si + , O + , and  − ) 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.

B. Nonlinear Parameter 𝛽 in Van Der Pol Equation
The dynamical behavior of the perturbed density  1 resulting from fiber-fuse propagation can be represented by the Van Der Pol equation: This means that the nonlinear parameter  determines the maximun amplitude of  1 .In the calculation, we used  = 6.5, which corresponds to   ≃ 0.8.

Figure 1 :
Figure 1: Schematic view of damaged optical fiber.

Figure 2 :
Figure 2: Temperature distribution of the high-temperature front versus the length along the  direction at  = 2 ms after the incidence of 1.8 W laser light ( 0 = 1.48 m) for the initial attenuation IA = 8 dB.The center of the high-temperature front is set at  = 0 m.

Figure 9
Figure 9: Relationship between the interval Λ and the input power  0 .The blue and black solid lines were calculated by using (6) and (4), respectively.The red open circles are the data reported by Todoroki[22,25].

Figure 10 :
Figure 10: Temperature dependence of the number densities of O and O + .

Figure 11 :
Figure 11: Temperature dependence of the production rate of O 2 .

Figure 12 :
Figure 12: Temperature distribution of the high-temperature front versus the length along the  direction at  = 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  = 0 m.