One of the extraordinary aspects of nonlinear wave evolution which has been observed as the spontaneous occurrence of astonishing and statistically extraordinary amplitude wave is called rogue wave. We show that the eigenvalues of the associated equation of nonlinear Schrödinger equation are almost constant in the vicinity of rogue wave and we validate that optical rogue waves are formed by the collision between quasi-solitons in anomalous dispersion fiber exhibiting weak third order dispersion.
© 2015 Optical Society of America
Rogue wave is a nonlinear wave phenomenon in which temporally and spatially localized wave with extraordinary amplitude that appears from a low-level average background. Rogue wave phenomenon in oceans generates a height of 30m huge wave in a short time. This has become one of the causes of maritime disasters damaging to ships or oil platforms. In recent years, research on rogue wave phenomenon has been actively carried out in various fields including light waves other than hydrodynamic context [1, 2]. The main characteristics of the temporally and spatially localized devastating freak waves appearing in the ocean can also be observed in supercontinuum generation in optical fiber. In supercontinuum generation, initial light in a narrow frequency band is converted into an ultra-broadband light. According to pioneering measurements by Solli et al , temporally and spatially localized extraordinary high amplitude peaks have been observed in average low-level background. They also pointed out that oceanic and optical rogue wave phenomenon is associated with the soliton phenomenon. Generally fundamental solitons are the nonlinear pulses which propagate without any distortion through a dispersive medium. In the anomalous dispersion fiber, higher-order soliton is generated from the continuous wave (CW) by modulational instability process. When considering the third order dispersion (TOD) of the fiber, higher-order solitons split into multiple quasi-solitons which correspond to multiple eigenvalues of the eigenvalue equation associated to the nonlinear Schrödinger equation (NLSE), while they propagate in the fiber. It has been reported that the collision of quasi-solitons is one of the generation mechanisms of the rogue wave [3,4]. Quasi-soliton collision process will be mainly focused in this paper. Other than quasi-soliton collision process, collision of Akhmediev breather (AB)s , and optical turbulence  are also considered as the generation mechanisms of optical rogue waves. Soliton solutions, and ABs are the two well-known classes of solutions of the NLSE . Wave turbulence is a classical nonlinear phenomenon which is observed in a variety of physical systems . Soliton collsion process can be figured out by utilizing the eigenvalues of eigenvalue equation associated to NLSE, since other sources such as ABs, optical turbulence, and Peregrine soliton cannot be analyzed using eigenvalues. It is very difficult to analyze the relationship between TOD and other phenomenon except soliton collision due to the variation in experimental conditions.
Recent studies have confirmed that the presence of TOD in optical fibers turns the system convectively unstable and generates extraordinary optical intensities. This statistical signature has been experimentally observed and confirmed numerically both in hydrodynamics and optics [9,10]. The crucial role of TOD on the statistics of optical rogue waves will be emphasized in this paper. We numerically examine the relationship between the magnitude of TOD and the quasi-soliton collision process which is considered as one of the mechanisms of generating optical rogue waves. The quasi-soliton collision process will be validated by solving the eigenvalue equation associated to NLSE. Invariant property of the eigenvalues during the collision reveals that rogue wave is generated by the collision of two quasi-solitons.
This paper is organized as follows. In section 2, modulational instability concept and eigenvalue equation associated to NLSE will be presented as an introductory overview. Numerical simulation model will be shown and the impact of TOD on extraordinary waves as well as the rogue wave generation process will be demonstrated in section 3. We discuss the stability of quasi-soliton against the magnitude of TOD coefficient in section 4. We validate the magnitude of TOD coefficient with which the generation mechanism of optical rogue waves can be explained by quasi-soliton collsion process.
2. Principles of modulational instability and eigenvalue equation associated to NLSE
Many nonlinear systems exhibit an instability referred to modulational instability in anomalous dispersion regime and manifest itself as breakup of the CW or quasi-CW radiation into a train of ultra-short pulses as a result of interplay between the nonlinear and dispersive effects . The steady state stability depends critically on whether light experience a normal or anomalous group velocity dispersion (GVD) in the optical fiber. In the nonlinear theory of the hydrodynamic context, the waves grow exponentially due to Benjamin-Feir instability . Instability increases exponentially in the initial stage and reaches to some saturated level. After reaching to the saturated level, it returns to the initial stage gradually. In other words, in a conservative system, all waves appear from nowhere and always disappear without a trace. A number of theories can be considered for the formation of optical rare or unexpected events under different conditions. Modulational instability, a complex nonlinear process exhibiting emergent behaviour and a strong sensitivity to initial conditions [13,14], has been found to play a crucial role in the appearance of extraordinary waves in many optical scenarios , [14–16].
As shown schematically in Fig. 1, when we launched some perturbed CW light as the initial waveform into an anomalous dispersion fiber, where a localized wave is formed by the modulational instability after propagating some distance. Figure 2 shows a numerical simulation result in which a perturbed CW light transforms into higher-order soliton pulse train and a dispersive wave. When the TOD is zero, higher-order solitons generated by the modulational instability process, propagate through the fiber by changing the temporal waveform and frequency spectrum periodically. On the other hand, when considering the non-zero TOD case, higher-order solitons split into multiple quasi-solitons which correspond to multiple eigenvalues of the eigenvalue equation associated with NLSE while they propagate in the fiber.
The behavior of the complex envelope of a light wave propagating in a fiber in the presence of anomalous GVD, nonlinearity, and TOD can be expressed as,17].
When σ = 0, the eigenvalue equation associated to Eq. (1) can be represented as,18]. As long as u is a solution of Eq. (1) for σ = 0, eigenvalue ζ of Eq. (2) is invariant with Z.
In order to determine the eigenvalue ζ, Eq. (2) is converted to integral equations by performing Fourier transformation defined byEq. (2) is transformed to Eq. (4) can be converted into discrete form for a fixed distance. Equation (5) can be rewritten as an eigenvalue problem of matrix form as shown in Eq. (6).
First, we assumed the initial waveform given by Eq. (9), and show the invariance of imaginary part of the eigenvalues in soliton collision process when σ = 0.Fig. 3(a). In Fig. 3(a), extraordinary optical intensities are formed when two solitons are in a collision around Z = 10. Moreover, sum of amplitude of each soliton is equal to the amplitude at collision point, which is 2 + 2 = 4. When two solitons are being in a collision, their eigenvalues are invariant before, during, and after the collision as shown in Fig. 3(b). We note here that two times imaginary part of observed two eigenvalues are close two the amplitude of initially launched pulses, i.e. A = 2. Eigenvalue of Eq. (2) can only be considered for weak TOD coefficient in Eq. (1) or weak Raman scattering coefficient in the Raman scattering term applied extended NLSE. Therefore, we restricted our numerical analysis for weak TOD in this paper. We will discuss the eigenvalue of Eq. (2) for large TOD coefficients in our further research.
3. Numerical simulation model and results
In this section, split-step Fourier method has been used to solve the NLSE (1), in order to numerically simulate the rogue wave phenomenon. Here W = 20 was taken as the width of the time window. The number of sampling points in the time window was set as 211. ΔZ = 10−5 is the step size to Z-direction. We defined the energy , and then we set the average power E/W = 1. A perturbed CW light was used as the initial waveform as shown in Fig. 4. In order to perturb the CW light, Gaussian noise with 10−5 of normalized average power was used. The rogue wave phenomenon was investigated by propagating the initial waveform until Z = 2000. Rogue wave profile, contour plot, and the variations of imaginary part of eigenvalues 2Im [ζ] with respect to Z in the vicinity of observed rogue wave were investigated.
First, we evaluated the impact of TOD on optical rogue waves. We described the rogue wave phenomenon by excluding the TOD coefficient in Eq. (1) as the initial stage of our analysis. Figure 5(a) illustrates the temporal waveform with respect to Z in the vicinity of maximum peak power extraordinary wave. Rogue wave event was identified at the distance of Z = 827.47. We could clearly observe an extraordinary optical intensity when compared with other waves at surroundings in Fig. 5(a). The mean height of the highest third of detected optical intensities is called the significant wave height (SWH) . Peak power of the observed extraordinary localized wave at the distance of Z = 827.47 is more than two times of SWH. Therefore, the localized structure emerging in chaotic wave field is very clearly can be described as a rogue wave event. The contour plot in the vicinity of optical rogue wave has been illustrated in Fig. 5(b). In addition, variations of imaginary part of eigenvalues 2Im [ζ] which was calculated by using temporal waveform have shown in Fig. 5(c) with respect to Z. Imaginary part of the eigenvalues have been almost preserved in the vicinity of Z = 827.47 and it can be regarded as collision of two solitons corresponding to two eigenvalues that generate the rogue wave. Imaginary part of the eigenvalues are 2.1 and 1.8, hence the sum of them is 3.9. On the other hand, the observed maximum peak power is 13.0, and the corresponding amplitude is which is approximately equal to 3.9, which is the sum of the amplitude of two solitons. Therefore, it is reasonable to state that the rogue wave is generated by soliton collision in this case. Moreover, soliton collision can be clearly observed in contour plot in the vicinity of optical rogue wave generation. The maximum peak power of the generated rogue wave was 13 times higher than the average power of the initial waveform when σ = 0.
By following a similar procedure and simulation parameters used for σ = 0, we analyzed the rogue wave phenomenon for σ ≠ 0. For a precisely quantitative evaluation we altered the magnitude of TOD finely and considered the relationship between the maximum peak power wave when initial waveform propagates until Z = 2000. Numerical simulations were performed for |σ| < 0.03. As shown by blue line in Fig. 6 maximum achieved peak power of extraordinary wave depends on the magnitude of TOD coefficient and the peak is maximized around |σ| ≃ 0.02. The maximum achieved peak power of the optical rogue wave was 61.7 when σ = −0.0186. For |σ| < 0.02, the maximum achieved peak power increased with |σ|. On the other hand, maximum achieved peak power decreased when 0.02 < |σ| < 0.03. Moreover, the maximum peak power does not depend on the sign of the TOD, and depends only on the absolute value of TOD. This tendency has given an interest to analyze the optical rogue wave phenomenon more precisely. In order to analyze the optical rogue wave phenomenon in detail, we mainly focused on (a) σ = 0.015 and (b) σ = 0.02 situations. We illustrated the wave profile, contour plot, and variations of imaginary part of the eigenvalues in the vicinity of maximum peak power extraordinary wave when the initial wave propagates until Z = 2000.
3.1. σ = 0.015 case
Here we demonstrate the rogue wave phenomenon for σ = 0.015. The observed wave profile in the vicinity of maximum peak power extraordinary wave is shown in Fig. 7(a). Figure 7(b) illustrates the contour plot in the vicinity of extraordinary wave generation. The maximum achieved peak power at Z = 1706.37 was 22.0 and it is slightly large comparing with σ = 0 situation. The variations of imaginary part of eigenvalues from the beginning to the end of optical rogue wave generation process with respect to Z is shown in Fig. 7(c). The imaginary part of two eigenvalues have been almost preserved in the vicinity of Z = 1706.37 and those are quasi-solitons corresponding to two eigenvalues 2.5 and 1.8. Therefore, the sum of each amplitude of two quasi-solitons is 2.5 + 1.8 = 4.3. Meanwhile, peak power of the emerged optical rogue wave is 22 and the corresponding amplitude is . Therefore, we can conclude that those values are almost the same and the rogue wave is generated by collision between two quasi-solitons. Moreover, we can clearly observe soliton collision in contour plot in the vicinity of optical rogue wave generation process.
3.2. σ = 0.02 case
Here we demonstrate the extraordinary wave generation for σ = 0.02 situation. Figure 8(a) illustrates the wave profile in the vicinity of highly localized wave. Optical rogue wave event has been identified at the distance of Z = 1885.75 when compared with other waves at background. Contour plot in the vicinity of optical rogue wave generation is illustrated in Fig. 8(b). The variations of imaginary part of eigenvalues 2Im [ζ] in the vicinity of Z = 1885.75 are shown in Fig. 8(c). The maximum achieved peak power at Z = 1885.75 is 59.1 and it is extremely large comparing with σ = 0 case. If we assume that this rogue wave was generated by collision between two quasi-solitons, the imaginary part of two eigenvalues are 6.1, and 0.1 as shown in Fig. 8(c). Therefore, the sum of them is 6.1 + 0.1 = 6.2. Meanwhile, amplitude of the emerged optical rogue wave is which is a quite different value. Moreover, observation of quasi-soliton collision is quite difficult in the vicinity of optical rogue wave generation, and the propagation of only one quasi-soliton could be seen in contour plot. Therefore, quasi-soliton collision can not be considered as the generation mechanism of the optical rogue wave in this context.
According to numerical simulation results shown in subsections 3.1 and 3.2, rogue wave can be generated due to different mechanisms depending on the magnitude of TOD. In the following, we focus on the rogue wave generated by quasi-soliton collision process. Numerical simulations have been performed for |σ| < 0.03, under the same conditions and parameters as the previous analysis. Figure 9 shows the imaginary part of two largest eigenvalues of the observed rogue wave. For |σ| < 0.015, two similar eigenvalues were observed. On the other hand, for |σ| > 0.02, the largest eigenvalue is much larger than the second largest one. 0.015 < |σ| < 0.02 is the transition region of the above two regions.
Red line in Fig. 6 shows the calculated maximum peak power from the imaginary part of eigenvalues shown in Fig. 9 under the assumption of quasi-soliton collision generating rogue wave. For |σ| < 0.015, calculated values agree with the maximum peak power achieved in the numerical simulation. Therefore, it is reasonable to consider the quasi-soliton collision as the generation mechanism of rogue waves in this parameter region.
4. Stability of quasi-solitons
Pure soliton solution cannot exist when σ ≠ 0 in Eq. (1). Weak TOD for |σ| ≪ 1 case can be considered as the perturbation to pure soliton and the soliton which behaves like a solitary wave is called quasi-soliton. While eigenvalue ζ in Eq. (2) is invariant for u which satisfies the NLSE in the case of σ = 0, the invariance of eigenvalue is not guaranteed mathematically for σ ≠ 0 situation. So, we examine the relationship between the stability of quasi-soliton and the magnitude of TOD. We have used a soliton pulse u(0, T) = η sech(ηT) as the initial waveform and investigated the variation of the imaginary part of eigenvalues when the pulse propagates until Z = 20 for the σ ≠ 0 situation. Figure 10 summarizes the results. When the imaginary part of the eigenvalue does not change more than 15% of the initial value, we consider that the quasi-soliton can propagate stably and we denote the parameter by circles in Fig. 10. On the other hand, 15% exceeding cases are considered as unstable and denoted by crosses in Fig. 10.
For |σ| < 0.03, the variation of the eigenvalue is small when the amplitude of initial pulse which corresponds to the imaginary part of eigenvalue is less than 5. As shown in Fig. 9, the imaginary part of the maximum eigenvalue of colliding quasi-soliton is around 6 for 0.02 < |σ| < 0.03 and it is unstable in Fig. 10. Therefore, the generation mechanism of rogue wave cannot be explained by the quasi-soliton collision process for |σ| > 0.02 case. On the other hand, for |σ| < 0.015, when the imaginary part of the eigenvalue becomes less than 5 as shown in Fig. 9 and the quasi-soliton is stable as shown in Fig. 10. Therefore, quasi-soliton collision can be considered as the generation mechanism of optical rogue waves for |σ| < 0.015. As we have already mentioned above, 0.015 < |σ| < 0.02 is the transition region of the above two regions.
In this paper, we have numerically analyzed the rogue wave phenomenon in optical fiber exhibiting weak TOD. By comparing the observed maximum amplitude of rogue wave and the sum of the two largest eigenvalues obtained by solving the eigenvalue equation associated with NLSE, we have shown the quasi-soliton collision process is the generation mechanism of rogue wave for weak TOD situation. For large TOD context, we need further studies to explain the generation mechanism of the rogue wave. Moreover, a strong influence of Raman scattering on formation of optical rogue waves was also identified which was similar to TOD. We will investigate the influence of Raman scattering on soliton collision process in the same framework in the future studies.
This work was partially supported by λ-reach project conducted by National Institute of Information and Communications Technology, Japan.
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