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
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
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].
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
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
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
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
where
The angular frequency
where
where
The oscillatory motion for
On the other hand, it can be seen that for
Next, the oscillatory motion for
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
In Figure 7,
The relationship between the period
where
Figure 8 shows the relationship between
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
2.1 Power dependence of periodic cavity interval
It is well known that the fiber-fuse propagation velocity
In this study the author investigated the
To explain the experimental
where
The second term
On the other hand, the relationship between the nonlinearity parameter
where
Using Eq. (5),
As shown in Figure 9,
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
3. Effect of nonlinearity parameters on cavity patterns
The nonlinearity parameter
The
Next, the oscillatory motion for
Figure 12 shows the relationship between
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
Todoroki reported that
We consider the tensile stress
On the other hand, it is well known for various solid materials that the
By using Eq. (11) and
The excess volume
As the maximum elongation rate
On the other hand, the volume
It is considered that the volume required to generate a cavity was compensated by the excess volume
Rearranging Eq. (14), we obtain the following inequality for
When
When
However, as shown in Figure 12,
As shown in Eq. (15), the allowable value of
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
3.2 Oxygen production in optical Fiber
When gaseous SiO and/or
In a confined core zone, and thus at high pressures,
The number densities
The dependence of
The number density
where
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 (
The rate equation of this reaction is [70]
where
The dependence of
Figure 18 shows the temperature distribution of the high-temperature front along the
This
It is expected that the
When
3.3 Diffusion length of oxygen gas
The
The diffusion coefficient
where
The mean square of the displacement
The
This
Figure 20 shows a schematic view of the diffusion of the
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
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 (
In a confined core zone, and thus at a high pressure,
If thermally produced electrons in the plasma are not bound to positive species (
Some electrons arrive at the surface of melted silica glass, and they attach to oxygen atoms on the surface because oxygen atoms have a high electron affinity [78]. As a result, a negatively charged surface, which was proposed by Yakovlenko [33], may be formed as shown in Figure 21.
However, the negative charges on the surface will immediately be balanced by an equal number of oppositely charged
The thickness
where
A cross section of the high-temperature plasma in the optical fiber with the double layers is schematically shown in Figure 22.
In the central domain of the high-temperature plasma, electrically neutral atoms (Si and O) and charged species (
The dynamical behavior of the perturbed density
where
If the solution of Eq. (26) is written as
where the amplitude
Differentiating Eq. (28), we obtain
Because of the slowly varying property of
The maximum value of
This means that the nonlinearity parameter
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