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.
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 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 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 by using a low-crosstalk 12-core fiber [5].
The fiber-fuse phenomenon was first observed in 1987 by British scientists [6–9]. Several review articles [10–14] 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]. 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 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 dcr, which is usually larger than the core diameter 2 rc, is a characteristic dimensional parameter of the fiber fuse effect. In an inner area with diameter d≤dcr, a fiber fuse (high-temperature plasma) propagates and silica glass is melted [17]. dcr, 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 d≤dcr 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 [20–28] (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 [26–28].
Schematic view of damaged optical fiber.
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 bullet-shaped cavities were observed by injecting pulses with 190 ps FWHM [7]. Dianov and coworkers observed the formation of periodic bullet-shaped cavities 20–70 μs after the passage of a plasma leading edge [29, 30]. Todoroki classified fiber-fuse 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 ω0 [34–52].
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.
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 frequency ω0 of the oscillation of the gas plasma is determined by the fiber-fuse propagation velocity Vf and the free-running distance Lf of the oscillator in the front region of the plasma and is given by(1)ω0=2π2VfLf.
The plasma instability is enhanced in the vicinity of the high-temperature peak. Therefore, in this study, the author estimated Lf 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 L along the z 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 α of 1.84 ×106m-1 when the thickness of the absorption layer, which consists of carbon black, is about 1 μm [54]. As shown in Figure 2, Lf was estimated to be about 12.5 μm, where Vf = 1 m/s was used in the calculation.
Temperature distribution of the high-temperature front versus the length along the z direction at t = 2 ms after the incidence of 1.8 W laser light (λ0 = 1.48 μm) for the initial attenuation IA = 8 dB. The center of the high-temperature front is set at L = 0 μm.
Here, the density ρ is considered in the form ρ=ρ0+ρ1, where ρ0 is the initial density of the stationary (unperturbed) part in the front region of the plasma and ρ1 is the perturbed density. The dynamical behavior of ρ1 resulting from fiber-fuse propagation can be represented by the Van Der Pol equation:(2)d2ρ1dt2-ϵ1-βρ12dρ1dt+ω02ρ1=0,where ϵ is a parameter that characterizes the degree of nonlinearity and β characterizes the nonlinear saturation (see Appendix B).
The oscillatory motion for ϵ = 0.1 and β = 6.5 is shown in Figure 3, where the perturbed density ρ1 is plotted as a function of time. When t≥ 80 μs, the maximum and minimum values of ρ1 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].
Time dependence of the perturbed density during fiber-fuse propagation. ϵ = 0.1; β = 6.5.
On the other hand, it can be seen that for ϵ = 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 μs.
Next, the oscillatory motion for ϵ = 5, 9, and 14 with β=6.5 was examined. The calculated results are shown in Figures 4–6, respectively.
Time dependence of the perturbed density during fiber-fuse propagation. ϵ = 5; β = 6.5.
Time dependence of the perturbed density during fiber-fuse propagation. ϵ = 9; β = 6.5.
Time dependence of the perturbed density during fiber-fuse propagation. ϵ = 14; β = 6.5.
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 [53], and the term “self-excited oscillation” is also often used.
The Φ values 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 Φ values are much larger than those for ϵ = 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 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.
Schematic view of the dimensional relationship between the temperature and the perturbed density of the ionized gas plasma during fiber-fuse propagation.
The relationship between the period Φ and the interval Λ is(3)Λ=ΦVf.The Λ values 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 (3) and Vf = 1 m/s. If a large amount of molecular oxygen (O2) 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 μm) are close to the experimental periodic void intervals of 13–22 μm observed in fiber-fuse propagation [22, 55].
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 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.
Relationship between the period Φ and the nonlinearity parameter ϵ. β = 6.5.
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 P0 observed in fiber-fuse propagation.
2.1. Power Dependence of Periodic Void Interval
It is well known that the fiber-fuse propagation velocity Vf increases with increasing the input laser power P0 [7, 8, 21, 22, 24, 25, 56–58]. Furthermore, in addition to Vf, Todoroki reported the P0 dependence of Λ in a SMF-28e fiber at λ0 = 1.48 μm [22, 55].
In this study, the author investigated the P0 dependence of Λ using the experimental Vf values [22, 25] and the calculated Φ values shown in Figure 8.
To explain the experimental Λ values in the P0 range from the threshold power (Pth≃ 1.3 W [59]) to 9 W, Λ(P0) can be represented by the following equation:(4)ΛP0=Φ0VfP01-γΦnϵ-Φnϵ=0Φ0,where Φ0 and γ are constants and Φn is the calculated Φ value shown in Figure 8. The second term -γΦn(ϵ)-Φn(ϵ=0)Vf(P0) 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 ϵ and P0 can be expressed as follows:(5)ϵ=χP0-Pthm/2,where χ is a constant and m is the order of the square root of the power difference P0-Pth. ϵ and χ correspond to the induced polarization and nonlinear susceptibility in nonlinear optics, respectively [60]. In the calculation, the author adopted χ = 1 and m = 2.
By using (4), Φ0 = 31.5 μs, γ = 3.6, and the Φn 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 was the calculation curve using the following equation:(6)ΛP0=Φ0VfP0,which is the first term on the right-hand side of (4).
Relationship between the interval Λ and the input power P0. 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].
As shown in Figure 9, Λ exhibits a steep rise near the threshold power (Pth) and increases with increasing P0. The Λ values at P0 = 2.0–2.5 W obey (6). However, with increasing P0, the Λ values at P0 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 P0 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 P0 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 P0. 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 O2 gas in the high-temperature core layer are described.
2.2. Oxygen Production in Optical Fiber
When gaseous SiO and/or SiO2 molecules are heated to high temperatures of above 5,000 K, they decompose to form Si and O atoms and finally become Si+ and O+ ions and electrons in the ionized gas plasma state.
In a confined core zone, and thus at high pressures, SiO2 is decomposed with the evolution of SiO gas or Si and O atomic gases at elevated temperatures [61]:(7)SiO2⇄SiO+12O2⇄Si+2O.The number densities NSiO, NSi, and NO (in cm−3) can be estimated using the procedure described in [62, 63] and the published thermochemical data [64] for Si, SiO, O, O2, and SiO2.
The dependence of NO on the temperature T is shown in Figure 10. NO gradually approaches its maximum value (3.3 ×1021cm-3) 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:(8)O⇄O++e-.
Temperature dependence of the number densities of O and O+.
The number density NO+ of O+ ions can be estimated using the Saha equation [63, 65]:(9)NO+2NO≈22πmekT3/2h3Z+Z0exp-IpkBT,where Ip (= 13.61 eV [66]) is the ionization energy of a neutral O atom, me is the electron mass, h is Planck’s constant, and kB is Boltzmann’s constant. Z+ and Z0 are the partition functions of ionized atoms and neutral atoms, respectively, and Z+≈Z0. The relationship between NO+ and T is also shown in Figure 10. NO+ increases gradually at temperatures above 7,000 K and reaches 8.9×1021cm-3 at 2 ×104 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] or 5–10 atmospheres [19]) at room temperature. The molecular oxygen (O2) is produced from neutral O atoms as follows:(10)2O⟶O2.The rate equation of this reaction is [67](11)dNO2dt=2πσ28RTπMONO2exp-EaRT,where σ (= 1.5 Å) is half of the collision diameter, MO (=16.0×10-3 kg) is the atomic weight of O, and Ea is the activation energy. The bond energy (493.6 kJ/mol [68]) of oxygen was used for Ea.
The dependence of dNO2/dt on the temperature T is shown in Figure 11. The rate of O2 production dNO2/dt exhibits its maximum value (2.96 ×1031cm-3s-1) at 12,700 K. This means that the oxygen molecules are produced most effectively at 12,700 K.
Temperature dependence of the production rate of O2.
Figure 12 shows the temperature distribution of the high-temperature front along the z direction at t = 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 L = 0 μm. In Figure 12, ΔLs, which is about 36.5 μm, is the distance between the high-temperature peak (L = 0 μm) and the location with a temperature of 12,700 K.
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 ΔLs can be converted into the time lag Δτs from the passage of the high-temperature front as follows:(12)Δτs=ΔLsVf.It is expected that the O2 molecular gas in the ionized gas plasma will be observed most frequently after a time lag of Δτs from the passage of the high-temperature peak. If the produced O2 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 Vf = 1 m/s, the Δτs values were estimated at a time of t = 1.55–3 ms after the incidence of 1.8 W laser light for IA = 8 dB. The calculated Δτs values are plotted in Figure 13 as a function of t. The fiber-fuse phenomenon was initiated at t = 1.5 ms (see Figure 14 in [54]). As shown in Figure 13, Δτs increases rapidly with increasing t immediately after the fiber fuse is initiated and reaches a constant value (36.5 μs) at t> 1.65 ms. This value is in reasonable agreement with the experimental values (20–70 μs) reported by Dianov and coworkers [29, 30].
Δτs values versus t after the incidence of 1.8 W laser light for IA = 8 dB.
2.3. Diffusion Length of Oxygen Gas
The O2 gas produced near the high-temperature front diffuses from the compressed part into the rarefied part of the oscillatory variation during a small period Φ of 10–30 μs (see Figure 9).
The diffusion coefficient D of the O2 gas is given by [67](13)D=23πσ2NO2RTπMO2,where MO2 (=32.0×10-3 kg) is the molecular weight of O2 gas. As NO2 is smaller than NO/2, NO2≈NO/2 is assumed in the calculation.
The mean square of the displacement Δz¯2 along the z direction of the optical fiber can be estimated from D and time t as follows [69]:(14)Δz¯2=2Dt.
The Δz¯ values at T = 12,700 K were estimated using (13) and (14). When t = 20 μs, the calculated Δz¯ value is given by(15)Δz¯=±16.7μm.This Δz¯ value is of the same order as the observed periodic void interval (13–22 μm) [55].
Figure 14 shows a schematic view of the diffusion of the O2 gas from the compressed part into the rarefied part in the high-temperature plasma. If the absolute value of Δz¯ is larger than half of the interval Λ between the periodic rarefied parts, many of the O2 molecules produced in the compressed part can move into the rarefied part during the period Φ (10–30 μs) of the relaxation oscillation. This O2 gas will form temporary microscopic voids that can constitute the nuclei necessary for growth into macroscopic bubbles [70].
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 O2 gas near the high-temperature front.
3. 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 ω such that ω≫ω0, where ω0 is the frequency of the Van Der Pol oscillator. With increasing the drive amplitude, the system behaves as if the previously existing relaxation oscillation (ω0) was destroyed (or quenched) by the asynchronous action of this frequency (ω).
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 (ϵ, β) and the physical properties observed in the fiber-fuse experiments is unknown. Therefore, to clarify this relationship, further quantitative investigation is also necessary.
AppendixA. 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 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: (A.1)Si+O⇄Si++O++2e-.
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 1×104 K. At high charge densities, it is known that the positive and negative species diffuse at the same rate. This phenomenon, proposed by Schottky [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 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.
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 [75].
The thickness δ0 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 Ne and T as follows [73]:(A.2)κ2=2Nee2ε0kBT,where e is the charge of an electron and ε0 is the dielectric constant of vacuum. When T=1×104 K and Ne=2.2×1020cm-3. Using these values and (A.2), the thickness δ0 of the double layer at 1×104 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.
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.
B. Nonlinear Parameter β in Van Der Pol Equation
The dynamical behavior of the perturbed density ρ1 resulting from fiber-fuse propagation can be represented by the Van Der Pol equation:(B.1)ρ¨1-ϵ1-βρ12ρ˙1+ω02ρ1=0,where ρ¨1 = d2ρ1/dt2, ρ˙1 = dρ1/dt, and ϵ and β are the nonlinear parameters.
If the solution of (B.1) is written,(B.2)ρ1=Acosω0t+φ,where the amplitude A and phase φ are slowly varying functions, then the A obeys the following equation:(B.3)A2=ρ12+ρ˙1ω02.
Differentiating (B.3), we get(B.4)A˙=ρ˙1ω02Aρ¨1+ω02ρ1=ρ˙1ω02Aϵ1-βρ12ρ˙1=ϵω02Aρ˙12-ϵβω02Aρ12ρ˙12=ϵAsin2ω0t+φ-ϵβA3sin2ω0t+φcos2ω0t+φ=ϵ2A1-cos2ω0t+2φ-ϵβ8A31-cos4ω0t+4φ.
Because of the slowly varying property of A, the oscillatory terms Acos(2ω0t+2φ) and A3cos(4ω0t+4φ) on the right-hand side of (B.4) are averaged out every cycle and can be discarded [76], so that (B.4) is reduced to(B.5)A˙≃ϵ2A-ϵβ8A3≃ϵ2A1-β4A2.
The maximum value of A, Am, is obtained under the condition of A˙ = 0. To satisfy this condition, Am obeys(B.6)Am=2β.This means that the nonlinear parameter β determines the maximun amplitude of ρ1. In the calculation, we used β = 6.5, which corresponds to Am≃ 0.8.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
AsanoA.KobayashiT.YoshidaE.MiyamotoY.Ultra-high capacity optical transmission technologies for 100 Tbit/s optical transport networks201194240040810.1587/transcom.e94.b.4002-s2.0-79851503112NakazawaM.Evolution of EDFA from single-core to multi-core and related recent progress in optical communication201421686287410.1007/s10043-014-0139-12-s2.0-84912113989MoriokaT.New generation optical infrastructure technologies: ‘EXAT initiative’ towards 2020 and beyondProceedings of the 14th OptoElectronics and Communications Conference (OECC '09)July 2009Hong KongIEEE1210.1109/oecc.2009.52131982-s2.0-70350710808RichardsonD. J.FiniJ. M.NelsonL. E.Space-division multiplexing in optical fibres2013753543622-s2.0-8487728944010.1038/nphoton.2013.94TakaraH.AsanoA.KobayashiT.KubotaH.KawakamiH.MatsuuraA.MiyamotoY.AbeY.OnoH.ShikamaK.GotoY.TsujikawaK.SasakiY.IshidaI.TakenagaK.MatsuoS.SaitohK.KoshibaM.MoriokaT.Proceedings of the 1.01-Pb/s (12 SDM/222 WDM/456 Gb/s) crosstalk-managed transmission with 91.4-b/s/Hz aggregate spectral efficiencyEuropean Conference and Exhibition on Optical Communication, ECEOC 2012September 201210.1364/ECEOC.2012.Th.3.C.12-s2.0-84882665198KashyapR.BlowK. J.Observation of catastrophic self-propelled self-focusing in optical fibres1988241474910.1049/el:198800322-s2.0-0024280360KashyapR.Self-propelled self-focusing damage in optical fibres1988859866HandD. P.RussellP. S.Solitary thermal shock waves and optical damage in optical fibers: the fiber fuse198813976776910.1364/ol.13.000767HandD. P.RussellP. St. J.Soliton-like thermal shock-waves in optical fibers: origin of periodic damage tracksEuropean Conference on Optical Communication1988111114AndréP.RochaA.DominguesF.FacãoM.BernardesM. A. D.Thermal effects in optical fibres2011chapter 1, pp. 1–20Rijeka, CroatiaInTech10.5772/822TodorokiS.YasinM.HarunS. W.ArofH.Fiber fuse propagation behavior2012chapter 20Rijeka, CroatiaInTech55157010.5772/2429KashyapR.The fiber fuse-from a curious effect to a critical issue: a 25th year retrospective20132156422644110.1364/oe.21.0064222-s2.0-84875207665TodorokiS.2014TokyoSpringerNIMS MonographsShutoY.YasinM.ArofH.HarunS. W.Simulation of fiber fuse phenomenon in single-mode optical fibers2014chapter 5Rijeka, CroatiaInTech159197MR3243133TodorokiS.Quantitative evaluation of fiber fuse initiation probability in typical single-mode fibersProceedings of the Optical Fiber Communication Conference2015OSA Technical Digest, Optical Society of AmericaTodorokiS.-I.Quantitative evaluation of fiber fuse initiation with exposure to arc discharge provided by a fusion splicer20166article 2536610.1038/srep253662-s2.0-84966293824DianovE. M.BufetovI. A.FrolovA. A.Destruction of silica fiber cladding by the fuse effect200429161852185410.1364/ol.29.0018522-s2.0-4043136446KurokawaK.HanzawaN.Fiber fuse propagation and its suppression in hole-assisted fibers201194238439110.1587/transcom.e94.b.3842-s2.0-79851505328DianovE. M.MashinskyV. M.MyzinaV. A.SidorinY. S.StreltsovA. M.ChickoliniA. V.Change of refractive index profile in the process of laser-induced fibre damage1992242932992-s2.0-0026944940DavisD. D.MettlerS. C.DiGiovanniD. J.Experimental investigation of the fiber fuse1995271420221010.1364/OFC.1995.WP17DavisD. D.MettlerS. C.DiGiovanniD. J.A comparative evaluation of fiber fuse models1996296659260610.1117/12.2742202-s2.0-0010531617TodorokiS.-I.Origin of periodic void formation during fiber fuse20051317638163892-s2.0-2414450078610.1364/OPEX.13.006381TodorokiS.-I.In situ observation of modulated light emission of fiber fuse synchronized with void train over hetero-core splice point200839, article e327610.1371/journal.pone.00032762-s2.0-54749120617DominguesF.FriasA. R.AntunesP.SousaA. O. P.FerreiraR. A. S.AndréP. S.Observation of fuse effect discharge zone nonlinear velocity regime in erbium-doped fibres20124820129512962-s2.0-8486708675210.1049/el.2012.2917TodorokiS.Fiber fuse propagation modes in typical single-mode fibersProceedings of the Optical Fiber Communication Conference2013OSA Technical Digest, Optical Society of AmericaAntunesP. F. C.DominguesM. F. F.AlbertoN. J.AndréP. S.Optical fiber microcavity strain sensors produced by the catastrophic fuse effect201426178812-s2.0-8489161431910.1109/LPT.2013.2288930LinG.-R.BaiadM. D.GagneM.LiuW.-F.KashyapR.Harnessing the fiber fuse for sensing applications20142288962896910.1364/oe.22.0089622-s2.0-84898949706De Fátima F.dominguesM. F. F.Brito PaixãoT. B.MesquitaE. F. T.AlbertoN.FriasA. R.FerreiraR. A. S.VarumH.Da Costa AntunesP. F. C.AndréP. S.Liquid hydrostatic pressure optical sensor based on micro-cavity produced by the catastrophic fuse effect201515105654565810.1109/JSEN.2015.24465342-s2.0-84939513287BufetovI. A.FrolovA. A.DianovE. M.FrotovV. E.EfremovV. P.Dynamics of fiber fuse propagationProceedings of the Optical Fiber Communication Conference2005OSA Technical Digest of OFC/NFOEC 2005, Optical Society of AmericaDianovE. M.FortovV. E.BufetovI. A.EfremovV. P.RakitinA. E.MelkumovM. A.KulishM. I.FrolovA. A.High-speed photography, spectra, and temperature of optical discharge in silica-based fiberss20061867527542-s2.0-3364463823110.1109/LPT.2006.871110AtkinsR. M.SimpkinsP. G.YablonA. D.Track of a fiber fuse: a Rayleigh instability in optical waveguides2003281297497610.1364/OL.28.0009742-s2.0-0037634034YakovlenkoS. I.Plasma behind the front of a damage wave and the mechanism of laser-induced production of a chain of caverns in an optical fibre20043487657702-s2.0-994426335410.1070/QE2004v034n08ABEH002845YakovlenkoS. I.Mechanism for the void formation in the bright spot of a fiber fuse20061634744762-s2.0-3375090369110.1134/S1054660X0603008XAbramsR. H.Jr.YadlowskyE. J.LashinskyH.Periodic pulling and turbulence in a bounded plasma19692272752782-s2.0-001446517010.1103/PhysRevLett.22.275StixT. H.Finite-amplitude collisional drift waves196912362763910.1063/1.16925272-s2.0-0014479447KeenB. E.FletcherW. H. W.Suppression and enhancement of an ion-sound instability by nonlinear resonance effects in a plasma1969231476076310.1103/PhysRevLett.23.7602-s2.0-0014666134KeenB. E.FletcherW. H. W.Suppression of a plasma instability by the method of 'asynchronous quenching'197024413013410.1103/PhysRevLett.24.1302-s2.0-0014699301KeenB. E.Interpretation of experiments on feedback control of a 'drift- type' instability197024625926210.1103/PhysRevLett.24.2592-s2.0-0014937673ShutkoA. V.Finite amplitude ion acoustic waves in an unstable plasma1970302248251NakamuraY.Suppression of two-stream instability by beam modulation1970285131513212-s2.0-000486686810.1143/JPSJ.28.1315KeenB. E.FletcherW. H. W.Remote feedback stabilization of the ion-sound instability by a modulated source at the electron-cyclotron resonance frequency19702563503532-s2.0-3594903537810.1103/PhysRevLett.25.350KeenB. E.FletcherW. H. W.Measurement of growth rate, non-linear saturation coefficients, and mode-mode coupling coefficients of a 'Van der Pol' plasma instability1970312, article 3151868188510.1088/0022-3727/3/12/3152-s2.0-0042844490NakamuraY.Suppression and excitation of electron oscillation in a beam-plasma system19713112732792-s2.0-2944443747910.1143/JPSJ.31.273KeenB. E.FletcherW. H. W.Nonlinear plasma instability effects for subharmonic and harmonic forcing oscillations197251, article 02015216510.1088/0305-4470/5/1/0202-s2.0-36149071168BoswellR. W.ChristiansenP. J.SalterC. R.Non linear effects in an R.F. plasma197238267682-s2.0-4964914009910.1016/0375-9601(72)90489-6TavzesR.ČerčekM.Frequency entrainment of a drift instability by nonlinear effects in a plasma1973432991002-s2.0-003900141810.1016/0375-9601(73)90563-XDeNeefP.LashinskyH.Van der Pol model for unstable waves on a beam-plasma system19733117103910412-s2.0-034336610910.1103/PhysRevLett.31.1039KeenB. E.FletcherW. H. W.The ion-sound instability and its associated multi-mode phenomena1973614168416982-s2.0-3364559440710.1088/0022-3727/6/14/305MichelsenP.PécseliH. L.Juul RasmussenJ.SchrittwieserR.The current-driven, ion-acoustic instability in a collisionless plasma1979211, article 005617310.1088/0032-1028/21/1/0052-s2.0-11744321698BuragohainA.ChutiaJ.NakamuraY.Mode suppression and period-doubling cascade in a double-plasma device19921635-64254282-s2.0-000737122510.1016/0375-9601(92)90850-LGyergyekT.ČerčekM.JelićN.StanojevićM.Mode suppression of a two-dimensional potential relaxation instability in a weakly magnetized discharge plasma1993177154602-s2.0-000483092310.1016/0375-9601(93)90373-8KlingerT.PielA.SeddighiF.WilkeC.Van der Pol dynamics of ionization waves19931822-33123182-s2.0-000023091810.1016/0375-9601(93)91079-KVan Der PolB.Technical papers: the nonlinear theory of electric oscillations19342291051108610.1109/JRPROC.1934.2267812-s2.0-0042630246ShutoY.End face damage and fiber fuse phenomena in single-mode fiber-optic connectors2016201611278139210.1155/2016/2781392TodorokiS.2014chapter 3Tokyo, JapanSpringerNIMS Monographs10.1007/978-4-431-54577-4_1ShutoY.YanagiS.AsakawaS.KobayashiM.NagaseR.Fiber fuse phenomenon in step-index single-mode optical fibers2004408111311212-s2.0-404312188710.1109/JQE.2004.831635FacãoM.RochaA. M.De Brito AndréP. S.Traveling solutions of the fuse effect in optical fibers201129110911410.1109/JLT.2010.20946022-s2.0-79551539812BumarinE. D.YakovlenkoS. I.Temperature distribution in the bright spot of the optical discharge in an optical fiber2006168123512412-s2.0-3375092459910.1134/S1054660X06080123TodorokiS.-I.Threshold power reduction of fiber fuse propagation through a white tight-buffered single-mode optical fiber2011823197819822-s2.0-8345518827310.1587/elex.8.1978BoydR. W.1992chapter 1New York. NY, USAAcademic PressMR2475397SchickH. L.A thermodynamic analysis of the high-temperature vaporization properties of silica19606043313622-s2.0-3384729281210.1021/cr60206a002ShutoY.Heat conduction modeling of fiber fuse in single-mode optical fibers201420141164520710.1155/2014/645207ShutoY.Evaluation of high-temperature absorption coefficients of ionized gas plasmas in optical fibers20102231341362-s2.0-7544911672810.1109/LPT.2009.203658919712ndU.S. Department of Commerce and National Bureau of Standards10.6028/NBS.NSRDS.37SahaM. N.Ionization in solar chromosphere19204023847248810.1080/14786441008636148KittelC.19967thchapter 3New York, NY, USAJohn Wiley & Sons, Inc.CastellanG. W.19833rdchapter 33BostonAddison-Wesley Publishing Co., Inc.19843rdTokyoChemical Society of JapanChap. 9,MooreW. J.19724thchapter 4PrincetonPrentice-Hall, Inc.BrennenC. E.2014chapter 1New York, NY, USACambridge University PressMR3155605SchottkyW.Diffusionstheorie der positiven Säule192425635640AllisW. P.RoseD. J.The transition from free to ambipolar diffusion195493184932-s2.0-000194020310.1103/PhysRev.93.84Zbl0055.23701PhelpsA. V.Diffusion of charged particles in collisional plasmas. Free and ambipolar diffusion at low and moderate pressures19909544074312-s2.0-002546432110.6028/jres.095.035OhtoriS.SekiguchiT.KawanoT.1969chapter 2Tokyo, JapanOhmsha Inc.IsraelachviliJ. N.19922ndchapter 12London, EnglandAcademic Press, Ltd.KawakuboT.KabashimaS.Stochastic processes in self-excited oscillation1974375119912032-s2.0-7795275984310.1143/JPSJ.37.1199