B. Nonlinearity parameter β in Van der pol equation

The dynamical behavior of the perturbed density ρ<sup>1</sup> resulting from fiber fuse propagation can be represented by the Van der Pol equation

$$
\ddot{\rho}\_1 - \varepsilon (\mathbf{1} - \beta \rho\_1^2) \dot{\rho}\_1 + a \rho\_0^2 \rho\_1 = \mathbf{0},\tag{26}
$$

where <sup>ρ</sup>€<sup>1</sup> <sup>¼</sup> <sup>d</sup><sup>2</sup> ρ1=dt<sup>2</sup> , ρ\_ <sup>1</sup> ¼ dρ1=dt, ε and β are nonlinearity parameters, and the nonlinearity parameter γ ¼ 0 is assumed.

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

$$
\rho\_1 = A \cos \left( a \rho\_0 t + \rho \right),
\tag{27}
$$

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

$$A^2 = \rho\_1^{-2} + \left(\frac{\dot{\rho}\_1}{a\rho\_0}\right)^2. \tag{28}$$

Differentiating Eq. (28), we obtain

$$\begin{split} \dot{A} &= \frac{\dot{\rho}\_1}{\omega\_0 2^2 A} \left( \ddot{\rho}\_1 + \omega\_0^2 \rho\_1 \right) \\ &= \frac{\dot{\rho}\_1}{\omega\_0 2^2 A} \left[ \varepsilon (1 - \beta \rho\_1^{-2}) \dot{\rho}\_1 \right] \\ &= \frac{\varepsilon}{\omega\_0 2^2 A} \left( \dot{\rho}\_1^{-2} \right) - \frac{\varepsilon \beta}{\omega\_0 2^2 A} \rho\_1^{-2} \left( \dot{\rho}\_1^{-2} \right) \\ &= \varepsilon A \sin^2 \left( a \omega\_0 t + \rho \right) - \varepsilon \beta A^3 \sin^2 \left( a \omega\_0 t + \rho \right) \cos^2 \left( a \omega\_0 t + \rho \right) \\ &= \frac{\varepsilon}{2} A \left[ 1 - \cos \left( 2a \omega\_0 t + 2\rho \right) \right] - \frac{\varepsilon \beta}{8} A^3 \left[ 1 - \cos \left( 4a \omega\_0 t + 4\rho \right) \right]. \end{split} \tag{29}$$

Because of the slowly varying property of A, the oscillatory terms <sup>A</sup> cos 2ð Þ <sup>ω</sup>0<sup>t</sup> <sup>þ</sup> <sup>2</sup><sup>φ</sup> and <sup>A</sup><sup>3</sup> cos 4ð Þ <sup>ω</sup>0<sup>t</sup> <sup>þ</sup> <sup>4</sup><sup>φ</sup> on the right of Eq. (29) are averaged out every cycle and can be discarded [80], thus reducing Eq. (29) to

$$\begin{split} \dot{A} &\simeq \frac{\varepsilon}{2}A - \frac{\varepsilon\beta}{8}A^3\\ &\simeq \frac{\varepsilon}{2}A\left(1 - \frac{\beta}{4}A^2\right). \end{split} \tag{30}$$

References

PDP5C.3

21(6):862-874

pilot tone. Optical Fiber

National Fiber Optic Engineers Conference 2012 (OFC/NFOEC2012);

[1] Sano A, Kobayashi T, Yamanaka S, Matsuura A, Kawakami H, Miyamoto Y, et al. 102.3-Tb/s (224 x 548-Gb/s) Cand extended L-band all-Raman transmission over 240 km using PDM-64QAM single carrier FDM with digital

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

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

[9] Hand DP, Russell PSJ. Soliton-like thermal shock-waves in optical fibers: Origin of periodic damage tracks. European Conference on Optical Communications. 1988:111-114

[10] André P, Rocha A, Domingues F, Facão M. Thermal effects in optical fibres. In: Bernardes MAD, editor. Developments in Heat Transfer. Rijeka,

[11] Todoroki S. Fiber fuse propagation behavior. In: Moh Y, Harun SW, Arof H, editors. Selected Topics on Optical Fiber Technology. Rijeka, Croatia: InTech;

[12] Kashyap R. The fiber fuse–From a curious effect to a critical issue: A 25th year retrospective. Optics Express. 2013;

[13] Todoroki S. Fiber Fuse: Light-Induced Continuous Breakdown of Silica Glass Optical Fiber. NIMS Monographs. Tokyo: Springer; 2014

[14] Shuto Y. Simulation of fiber fuse phenomenon in single-mode optical fibers. In: Yasin M, Arof H, Harun SW, editors. Advances in Optical Fiber Technology. Rijeka, Croatia: InTech;

[15] Kashyap R. High average power effects in optical fibres and devices. In: Limberger HG, Matthewson MJ, editors. Proceedings of Society of Photo-Optical Instrumentation Engineers. 2003. Vol.

[16] Todoroki S. Quantitative evaluation of fiber fuse initiation probability in typical single-mode fibers. Optical Fiber Communication Conference; 2015

[17] Todoroki S. Quantitative evaluation of fiber fuse initiation with exposure to arc discharge provided by a fusion

Croatia: InTech; 2011

21(5):6422-6441

2012

2014

4940. pp. 108-117

(OFC2015); W2A.33

Communication Conference Exhibition/

[2] Nakazawa M. Evolution of EDFA from single-core to multi-core and related recent progress in optical communication. Optical Review. 2014;

[3] Morioka T. New generation optical infrastructure technologies: "EXAT initiative" toward 2020 and beyond. OptoElectronics and Communications Conference 2009 (OECC 2009); FT4

[4] Richardson DJ, Fini JM, Nelson LE. Space-division multiplexing in optical fibres. Nature Photonics. 2013;7:354-362

[5] Takara H, Asano A, Kobayashi T, Kubota H, Kawakami H, Matsuura A, et al. 1.01-Pb/s (12 SDM/222 WDM/456 Gb/s) crosstalk-managed transmission with 91.4-b/s/Hz aggregate spectral efficiency. European Conference on Optical Communication 2012 (ECOC2012); Th.3.C.1

[6] Kashyap R, Blow KJ. Observation of catastrophic self-propelled self-focusing in optical fibres. Electronics Letters.

[7] Kashyap R. Self-propelled selffocusing damage in optical fibres. Proceedings of Xth International

[8] Hand DP, Russell PSJ. Solitary thermal shock waves and optical damage in optical fibers: The fiber fuse. Optics Letters. 1988;13(9):767-769

Conference on Lasers; 1988. pp. 859-866

1988;24(1):47-49

61

The maximum value of <sup>A</sup>, Am, is obtained under the condition of <sup>A</sup>\_ <sup>¼</sup> 0. To satisfy this condition,

$$A\_m = \frac{2}{\sqrt{\beta}}.\tag{31}$$

This means that the nonlinearity parameter β determines the maximum and minimum values of ρ1. In the calculation, we used β ¼ 6:5, which corresponds to Am ≃0:8.
