## Abstract

We have investigated and analyzed the modulation instability (MI) of finite energy Airy pulse (FEAP) in an optical fiber in order to reveal the impact of truncation coefficient on the nonlinear propagation dynamics of FEAP with or without amplitude perturbation. We have also characterized the difference between the propagation process of smooth FEAP and that of modulated FEAP. It is shown that, for a smooth FEAP, the side lobes prior to the main lobe first undergo compression and then break up into multiple sub-pulses during propagation in the case of small truncation coefficient; while the opposite occurs in the case of large truncation coefficient. For a FEAP with amplitude modulation, the breakup of the main lobe induced by MI precedes that of side lobes for arbitrary values of truncation coefficients; but the evolution of secondary lobes is made by a transition from splitting process to a simple compression process with increasing truncation coefficient. The propagation dynamics of secondary lobes with number symbol larger 2, marked the secondary lobes starting number 1 from near to far according the distance between itself and the main lobe, is insensitive to the truncation coefficients variation in both cases. Finally, the MI gain spectra of FEAP with different truncation coefficients are obtained by numerically solving the nonlinear Schrödinger equation and the results have been compared with the theoretical predictions.

© 2014 Optical Society of America

## 1. Introduction

Since the first observation of finite-energy Airy beam (FEAB) in 2007 [1, 2], there are much increasing attraction towards generation, manipulation and applications of the FEAB in both experimental and theoretical studies [3–6]. The FEAB is an extended model of Airy wave packet, which is based on a shape-preserving self-accelerating solution with the form of an Airy function to the Schrödinger equation originally found by Berry and Balazs [7]. Due to its intriguing characteristics of self-acceleration, quasi-diffraction-free and self-healing [1, 2, 8, 9], FEAB has been utilized for many different applications such as particle manipulation [10, 11], curved filament generation [12, 13], abrupt auto-focusing [14] and optical routing [15].

Recently, a new kind of optical pulse referred to as finite energy Airy pulse (FEAP), has been introduced in direct analogy with FEAB, because the temporal dispersive equation and the spatial diffraction equation are isomorphic [1]. Despite the fact that the mathematical description of Airy pulses is similar to that of Airy beams, there remains an important physical difference between temporal and spatial accelerations: an accelerating beam bends its trajectory in space, whereas the temporal accelerating pulse truly changes its actual group velocity. This has stimulated great interest for the problems of Airy pulses propagation from linear to nonlinear regimes as well as from temporal one-dimension to spatiotemporal three-dimension. In the linear regime, the propagation dynamics of FEAP has been reported [16–20]. In [16, 17], Besieris *et al.* and R. Driben *et al.* investigated the evolution of Airy pulses injected into a fiber close to its zero dispersion point in presence of dominant positive or negative third-order dispersion. FEAP has been used for synthesizing versatile linear light bullets. Chong *et al.* first reported the experimental demonstration of a versatile three-dimensional linear light bullet consisting of spatial Bessel beams and temporal Airy pulses [18]. Abdollahpour *et al.* realized Airy-Airy-Airy light bullets by combining an Airy pulse in time and Airy distribution in space [19]. These exciting results open up new possibilities in numerous applications and pave the way for the experimental realization of 3D light bullets interactions. FEAP propagating in the nonlinear regime was discussed as well [21–26]. Fattal *et al.* disclosed that the solitons will be shed from the main lobe of Airy pulses under combined effects of dispersion and Kerr nonlinearity [21]. The influence of Kerr nonlinearity on the evolution of FEAP in normal or anomalous dispersive regimes is disclosed [22, 23]. Ament *et al.* carried out an experimental and numerical investigation on supercontinnum generation by using a femtosecond FEAP in photonic crystal fiber [24]. Zhang *et al.* investigated the femtosecond self-decelerating FEAP propagating in an optical fiber with the inclusion higher-order linear and nonlinear effects such as Raman scattering, self-steepening as well as third-order dispersion [25]. Wang *et al*. carried out a study of the evolution dynamics of FEAP in optical fibers with periodic dispersion modulation [26].

It is well known that modulation instability (MI) is one of the most fundamental and universal effects related to wave propagation in nonlinear media. This nonlinear action can result in pattern formation from stochastic fluctuations in a wide variety of systems and has recently been regarded as ‘one of most ubiquitous types of instabilities in nature [27]. It is considered as a precursor to formation of localized nonlinear excitations, or soliton trains, exhibiting a close relation to the existence of both bright and dark solitons. In nonlinear optics, MI stems from the interaction between nonlinearity and dispersion (in temporal domain) or/and diffraction (in spatial domain), and can be classified as temporal [28, 29], spatial (or transverse) [30, 31] or spatiotemporal [32–34]. Apart from extensive investigations of MI in ordinary materials, MI has also been widely studied in metamaterials where the refractive index is negative in opposition to ordinary materials [35–39]. It is worth noting that MI theory is also used to interpret the formation of optical rogue wave recently observed in the context of a supercontinuum [40, 41] because it is the mechanism that initiates supercontinuum generation in the case of relatively long pulses. The spatiotemporal instability was used to explain the origin of the spatiotemporal filament dynamics of ultrashort pulses in nonlinear media [42, 43].

All those previous studies on MI mainly considered pump pulses (or beam) with symmetrical profiles, such as plane waves [28, 29], Townes profile [42], Bessel beam [43], or sech solutions [44]. The MI of the FEAP with a typical asymmetrical profile, to the best of our knowledge, remains however unexplored. Herein, we investigated MI of FEAP in nonlinear media with optical Kerr nonlinearity, and disclosed the novel characteristic of FEAP propagation in nonlinear Kerr medium. To study the propagating properties of FEAP in Kerr nonlinear medium, we first give a brief review of the MI theory. Then we present simulation results of MI behavior of FEAP, because it is not treated analytically. The propagation dynamics of FEAP with temporal modulation is also carried out through solving the NLSE numerically. In addition, the impact of the truncation coefficients of FEAP on the instability gain spectra of MI is disclosed as well.

## 2. Model of the nonlinear propagation of Airy pulse

In order to better investigate the nonlinear dynamics and MI, we use a simple model on the propagation of a laser pulse through an optical fiber. In the approximation of slowly varying amplitudes, the light field obeys the nonlinear Schrödinger equation (NLSE) written in the following normalized form

FEAP with the expression of $U\left(Z,T\right)=Airy\left(T\right)\mathrm{exp}\left(aT\right)$ is a solution to the Eq. (1) without the nonlinear effects. When FEAP propagates in optical fiber taking into account the self-phase modulation, a soliton will be shed from the main lobe owing to the combined effects of anomalous second-order dispersion and self-phase modulation [21]. One may ask about the evolution dynamics of FEAP, when the incident FEAP is perturbed by amplitude or phase noise. For the case of CW light with perturbation launched into optical fiber, the perturbation will experience nonlinear growth described by MI theory [28]. The growth rate $g$ is

Here ${\Omega}_{c}^{2}=4{N}^{2}$, $\Omega $ is the normalized frequency of perturbation with respect to the central frequency of input pulse ${\omega}_{0}=1/{t}_{0}$. When instability occurs, the fastest growth mode is most interesting. The fastest growth frequencies ${\Omega}_{\mathrm{max}}$ and the corresponding maximum gain ${g}_{\mathrm{max}}$ of this mode is ${\Omega}_{\mathrm{max}}={\Omega}_{c}/\sqrt{2}$ and ${g}_{\mathrm{max}}={\Omega}_{c}^{2}/2{L}_{D}$ respectively.## 3. Numerical results of modulation instability of Airy pulse

Now, let us focus our attention on the MI of Airy pulse. First, we study the propagation dynamics of Airy pulse with temporal amplitude modulation in Kerr nonlinear medium. The initial pumped field is FEAP with a cosinoidal modulation,

*a*. Parameter $\alpha $ is the initial amplitude of modulation wave, which is set to be 0.05; modulation frequency $\Omega $ is chosen such that it is the fastest growing frequency${\Omega}_{\mathrm{max}}$.

Figure 1 shows a comparison of the propagation of four FEAPs which have different modulation depths as well as different truncation coefficients. The widths of main lobe of four FEAPs are almost identical. The left column and right column of Fig. 1 display contour maps of temporal evolution of four FEAPs with modulation depth $\alpha =0$ (left column) and $\alpha =0.05$ (right column) for different truncation coefficients: the truncation coefficient is *a* = 0.01 in the first line, *a* = 0.05 in the second line, *a* = 0.10 in the third line and *a* = 0.15 in the fourth line. For the case of the modulation depth $\alpha =0$, it means that no amplitude modulation was imparted on the initial incident FEAP, whereas the propagation dynamics of such smooth FEAP shows some differences with the truncation coefficients increased. When the truncation coefficient *a* is set to 0.01, the compression of side lobes are stronger than that of main lobe in the first 2.3 propagation distance. With further increasing of the propagation distance, the main lobe continues to contract a fraction of its initial width, and then splits into multiple sub-pulses. The neighboring sub-pulses interact with each other. Such an interaction leads to exchange of energy between sub-pulses and stimulates new sub-pulse formation, making the propagation of the main lobe very complex. While the propagation dynamics of side lobes is relatively simple if compared to that of the main lobe; after reaching its maximum peak owing to the compression resulted from the combined effects of anomalous dispersion and nonlinearity, the side lobes will regularly split into two, three or four sub-pulses, depending on the initial intensity of side lobes and propagation distance. With further propagation, the FEAP eventually undergoes a reshaping into a multiple pulses pattern with multiple sub-pulses located at the positions of the main lobes of the input FEAP. When the truncation coefficient is *a* = 0.05, the temporal evolution of FEAP is similar to the case of *a* = 0.01. The number of side lobes experiencing splitting process decreases, and a few side lobes just split into two or three pulses [Fig. 1(b)]. Figures 1(c) and 1(d) show the propagation dynamics of FEAP for the case of strongly truncated degree, indicating that temporal evolution of lobes change significantly compared with the case of weakly truncated degree [Figs. 1(a) and 1(b)]. The number of sub-pulses split from the main lobe decrease, and the splitting process of side lobes is not obvious [Fig. 1(c)], and even does not occur [Fig. 2(d)]. When a cosine modulation is imposed on the initial incident FEAP, how does the temporal propagation dynamics progress? Figures 1(e)-1(h) show the temporal evolution of four modulated FEAPs as a function of propagation distance for different truncated degrees. It is shown that MI leads to a breakup of input pulses into a sub-pulse train, especially for the main lobe of FEAP. The impact of modulation on the splitting of side lobes grows smaller and smaller with increasing truncation coefficient *a*. Our simulations show that the value of truncation coefficient has an important effect on the nonlinear propagation of FEAP and on its deceleration or acceleration since the number of secondary lobes control the energy 〉ux that replenishes and pushes the main lobe.

We observe from numerical simulations a transition between a regime where the turning ability of the Airy pulse is preserved to a regime where the Airy peak trajectory is seemingly modified due to self-phase modulation of intense lobes and lobe competition for the available power. In the case of the small truncation coefficients, the linear energy 〉ux of the FEAP is sustained over a large propagating distance. FEAP with large truncation coefficient containing only 3 lobes propagates over a much shorter distance under the action of nonlinear effects. The linear energy 〉ux responsible for the curved trajectory in this case is not strong enough to bring energy from the tail to the peak of the FEAP during propagation. The larger truncation coefficient quenches the power replenishment of the secondary lobes and therefore simultaneously tends to quench the turning ability of the Airy pulse. The main lobe still undergoes self-phase modulation and form optical solitons, but they subsequently follow standard solitons interaction process featured by energy exchange with their neighborhood environment. These results seem highly correlated with the previous findings [45, 46].

One comparison between temporal evolution of the smooth FEAP (left column of Fig. 2) and that of modulated FEAP (right column of Fig. 1) reveals the main influence of MI on the main lobe of FEAP [Figs. 1(e)-1(h)], and its neighboring secondary lobes 1 and 2 (numbered starting 1 from near to far according to the distance between secondary lobes and the main lobe) [Figs. 1(e) and 1(f)]. The latter case appears strongly depending on the values of truncation coefficients. To intuitively compare the intensity profiles between the main lobe and side lobes, Fig. 2 shows the pulse shapes of smooth FEAP with the truncation coefficients *a* = 0.01 (left column) and *a* = 0.10 (right column) at output of different propagation distances. We define *the compression factor* as the ratio of output peak intensity to input peak intensity. At propagation distance *Z* = 2.3, the lobes contracts a fraction of its initial width, but the compression factor of main lobe is very small than that of secondary lobes for the case of small truncation coefficient (*a* = 0.01). The opposite case occurs for large truncation coefficient (*a* = 0.10). The compression factor of main lobe is about 1.7, but the secondary lobes exhibit high compression factor in the range 6 to 11. It is clearly shown in Fig. 2 that the side lobes earlier than the main lobe experience the process from compression to splitting for the case of weak truncated degree (*a* = 0.01). By contrast, the opposite occurs for the case of strong truncation degree (*a* = 0.10).

These new propagating behaviors of FEAP can be attributed to their intriguing characteristics of self-acceleration, quasi-dispersion-free and self-healing. For the case of weakly truncated degree, FEAP with a small truncation coefficient has more tails and is capable of representing the ideal Airy pulse. The characteristics of such FEAP are very approximately in agreement with that of ideal Airy pulses. Moreover, the self-healing behavior can postpone the compression and splitting of the main lobe. On the other hand, the property of self-deceleration is able to bend the sub-pulses, making a contribution to the interaction between sub-pulses. In addition, the peak intensity of side lobes becomes dramatically smaller with respect to that of main lobe as the truncation coefficient is increased. This can weaken the nonlinear effects.

Figure 3 shows the maximum intensity of smooth (dash-dot line) and modulated (solid line) FEAP as a function of propagation distance for different truncation coefficients (a) *a* = 0.01, (b) *a* = 0.05, (c) *a* = 0.10 and (d) *a* = 0.15. It is obviously displayed that the pulse compression and splitting are gradually delayed as the truncation coefficients increase. The maximum intensity evolutions have an oscillatory structure with multi-peaks, showing that sub-pulses interact with each other as well and exchange energy during this propagation process. In addition, the whole pulse compression coexist with the pulse splitting, so they exit together during the MI process and compete against each other.

It is difficult to obtain the expression of gain spectrum of FEAP in Kerr nonlinear medium. The gain spectrum will be obtained by numerically calculating the nonlinear growth of modulation. The gain is defined as

By applying amplitude modulation with different frequencies, modulation will experience different growth rate. Figure 4 shows the gain spectra of MI of FEAP with different truncation coefficients. It can be found from Fig. 4 that the fastest growing frequencies of gain spectra are basically identical for different truncation coefficients, and its maximum gain is inversely proportional to truncation coefficients. The gain spectra shapes are very similar. Surprisingly, the fastest growing frequency is located at about 0.9, which does not coincide with the theoretical value 1. The MI gain band of FEAP shrinks compared with that of theoretical predictions. The main reasons for this difference between theoretical results and numerical simulations can be interpreted as follows. The MI theory presented in Section 2 assumes that the input optical field is a CW light with infinite energy. The FEAP used in numerical calculation of gain spectra has not only an asymmetric multi-peak structure but also finite energy and intriguing characteristics of self-acceleration, quasi-diffraction-free and self-healing [1, 9].

## 4. Conclusions

In conclusion, we have numerically investigated the propagation dynamics of FEAP with and without temporal modulation. The influence of the truncation coefficient on the temporal evolution is discussed. It is shown that when the FEAP was not imposed a temporal modulation, its side lobes prior to the main lobe undergo compression, and split into multiple sub-pulses for the case of mall truncation coefficient, while the opposite occurs in the case of big truncation coefficient. As a cosine modulation is added to incident FEAP, the breakup of main lobe induced by MI is prior to that of secondary lobes for arbitrary values of truncation coefficients. However, the temporal evolution of secondary lobes is influenced by different values of truncated coefficient. Especially, the propagation dynamics of secondary lobes is made by a transition from compression-splitting process to a simply compression process with truncated coefficient increased. It is anticipated that these results may provide some useful and significant suggestions for the potential applications of FEAP in nonlinear optics.

## Acknowledgments

We thank Dr. Subhadeep Datta and Dr. Han Zhang for their support at improving the quality of English. This work is supported by the Natural Science Foundation of SZU (Grant Nos. 201449, 201450) and the Hunan Provincial Natural Science Foundation of China (Grant No. 13JJ4108).

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