## Abstract

We numerically simulate the propagation of finite energy Airy pulses in optical fibers with cubic-quintic nonlinearity and analyze the effects of quintic nonlinear parameters and soliton order number on their evolution properties. The soliton pulses are observed, whose peak amplitudes and corresponding temporal positions will vary with the propagation distance. Depending on different quintic nonlinearity parameters and soliton order number, the soliton pulse temporal positions exhibit weak decayed oscillations and then nearly linearly shift to leading or trailing edge of the Airy wavepacket, or tend to fixed positions, and the peak amplitudes also exhibit decayed oscillations but with different oscillation amplitude and central values. For large soliton order number, the soliton pulses are considerably compressed. Other weak dispersive wave pulses will appear near the main soliton pulses and gradually depart from the main soliton pulses. In the case of small soliton order, despite their considerable energy attenuation, the main lobes and even minority of the neighboring side lobes of the Airy pulses can still recover from the energy transfer to the soliton pulses and the dispersive wave pulses and maintain their unique properties of self-healing and self-acceleration in time for a very long distance. In the case of large soliton order, however, the Airy wavepacket only remains its very weak background and even disappears quickly.

© 2015 Optical Society of America

## 1. Introduction

Since their first predictions in quantum mechanics in 1979 [1] and especially the recent theoretical introductions and experimental demonstrations in optics in 2007 [2, 3 ], Airy wavepackets, which can be mathematically expressed as Airy function, have intrigued extensive research interests in optics, due to their unusual propagation features including self-bending (or self acceleration), self-healing, and weak diffraction, which allows for various exciting potential applications typically including optical trapping and micro-manipulation [4–6 ], light bullet generation [7, 8 ], curved plasma channel generation [9], vacuum electron acceleration [10, 11 ], supercontinuum generation [12], curved laser filamentation [13], optical routing [14], and etc. Researchers mainly pay attention to the experimental realization of optical Airy wavepackets [3, 9, 15 ], unique features during propagation, beam manipulation [15], various potential applications as just mentioned above, and so on. Now, the study of optical Airy wavepackets has been considerably extended from paraxial cases to nonparaxial ones [16–18 ], from spatial-domains to temporal-domains, from complete coherent fields to partially coherent ones [19–21 ], and so on. Moreover, the media involved have also developed from linear media to nonlinear ones, from ordinary media to Bose-Einstein condenstates [22], chiral media [23], the metal surface [24–28 ], photonic crystals [29], and atom vapors with electromagnetically induced transparency [30].

Besides the initially extensive study of spatial domain Airy wavepackets (Airy beams), the temporal domain Airy wavepackets, namely Airy pulses, have recently attracted tremendous research interests both in linear and nonlinear regimes [12, 31–37 ]. Airy pulses can be experimentally constructed by introducing cubic spectral phases in the Gaussian pulses [3]. Being quite analogous to their spatial domain counterparts, Airy pulses are highly dispersion resistant and self-healing in the linear dispersion materials [12, 31 ]. In the nonlinear regimes, Airy pulses are demonstrated to be more robust in adapting to the periodic dispersion modulation than sech-shaped pulses below a moderate pump power, beyond which they will collapse due to the detrimental effects of intense nonlinearity [32]. In particular, Airy pulses involved soliton sheding [33], inversion and tight focusing in temporal domain [34], modulation instability [35], supercontinuum generation [12], and propagation dynamics in case of higher-order effects [36, 37 ] such as Raman scattering, self-steepening, and third-order dispersion, are further investigated more recently. And more and more unusual and interesting nonlinear propagation properties for Airy pulses are revealed, which must be of theoretical and practical importance.

To our knowledge, present studies on nonlinear propagations of Airy pulses are only limited to the cubic nonlinearity case. However, as previous investigations have revealed, for high incident optical intensities or materials with very high nonlinear coefficients such as semiconductor doped glass fibers, quintic nonlinearity will take effect and influence considerably the optical soliton propagation [38], modulation instability [39], and optical wave breaking [40, 41 ]. That is to say, quintic nonlinearity has become so important that it is inappropriate to neglect it in this case. Naturally, as far as the nonlinear propagations of Airy pulses are concerned, people may ask whether Airy pulses can be robust enough to resist quintic nonlinearity effectively and continue remaining their unusual propagation features, or they will be influenced considerably and even collapse quickly with dispersive waves and solitons generation or other new nonlinear optical phenomena occurrence. Moreover, from the viewpoint of engineering applications, people pay more attention to the issue that whether they can exploit appropriate quintic nonlinearity to effectively manipulate the propagation properties and their corresponding applications of Airy pulses. To make all these things clear, quintic nonlinearity should generally be taken into account in the nonlinear propagation model. After all, the more general model can be utilized to obtain more believable research results in this case. So far, however, we have not seen reports on the propagation dynamics of Airy pulses in case of quintic nonlinearity. Thus, the purpose of this work is to numerically study and analyze the evolution dynamics of finite energy Airy pulses for different quintic nonlinearity parameters and soliton order number. The results indicate that, for large soliton order number, quintic nonlinearity can distort Airy pulses dramatically and lead to the weak dispersive waves and the intense compressed soliton generation. Different quintic nonlinearity parameters can influence the solitons in terms of their peak amplitudes and temporal positions in different ways. For small soliton order number, quintic nonlinearity only influences the Airy pulses propagation slightly. Accordingly, this work is of importance in enriching the investigation of Airy pulse related nonlinear propagation, soliton generation, and pulse compression. Furthermore, it may provide an alternative way to manipulate the soliton propagation by exploiting appropriate quintic nonlinearity.

## 2. Propagation model

In the presence of quintic nonlinearity, the extended nonlinear Schrödinger equation governing the optical field propagating in an optical fiber is of the following form [40, 41 ]

*z*the propagating distance, and

*T*the retarded time.

Generally, Eq. (1) is usually transformed to its normalized form in convenience for numerical simulations

*P*

_{0}and

*T*

_{0}are respectively the peak power and duration of the launched pulse. Sgn denotes the sign function. The normal dispersion region (${\beta}_{2}\text{\hspace{0.17em}}>\text{\hspace{0.17em}}0$) and the anomalous one (${\beta}_{2}\text{\hspace{0.17em}}<\text{\hspace{0.17em}}0$) correspond to $\mathrm{sgn}\text{\hspace{0.17em}}({\beta}_{2})\text{\hspace{0.17em}}\text{\hspace{0.17em}}>\text{\hspace{0.17em}}0$ and $\mathrm{sgn}\text{\hspace{0.17em}}({\beta}_{2})\text{\hspace{0.17em}}\text{\hspace{0.17em}}<\text{\hspace{0.17em}}0$ in Eq. (2), respectively. In this work, we focus on only the propagation property of Airy pulses in the anomalous dispersion region where bright optical solitons are supported.

The incident pulse is a finite energy Airy pulse with an exponentially decreasing function, which is thus experimentally realizable and can be written as

*b*is set to be 0.05 which is the truncation-dependent factor enabling the experimental realization of the Airy pulse, Ai represents the Airy function, and

*X*(

*b*) is introduced to keep the pulse amplitude to the soliton order

*N*for any value of

*b*.

## 3. Calculations and discussions

According to Eqs. (2) and (3)
, we can numerically simulate the evolution dynamics of the finite energy Airy pulse by employing the split-step Fourier algorithm. In order to explore effects of the peak power of the pulse and different quintic nonlinearity parameters on the pulse evolution, we will vary the values of the soliton order *N* and quintic nonlinearity related parameter *R* in the following simulations. Here, we respectively set the value of *N* to be l, 2, and 3. Besides, in order to observe more clearly the weak sub-pulses with lower intensity, what we display will be the evolution of the normalized amplitude │*u*│ rather than the normalized intensity │*u*│^{2} as Driben *et al*. did [34]. Figure 1
shows the contour maps of temporal evolution of finite energy Airy pulses for different values of soliton order *N* and quintic nonlinearity related parameter *R*. In Fig. 1, *Z* = *z* / *z*
_{0} represents another kind of normalized distance. In the calculation, we measure the propagation distance in *z*
_{0} units. For low value of *N*, which corresponds to low launched power, narrow pulse width, intense dispersion, or small cubic nonlinearity coefficient, the main lobes and even most of the side lobes of the Airy pulses can maintain their unique self-bent or self-acceleration properties in time, as shown in Figs. 1(a)-1(e) where *N* = 1. The only difference is that, on the background of the Airy waveform, an intense quasi-soliton pulse and a very weak dispersion wave will appear respectively near the pulse center and the leading edge. Furthermore, with increase of propagation distance, the former nearly remains near the pulse center whereas the latter gradually shifts towards the leading edge and disappears quickly. It can still be seen that the quintic nonlinearity only influences the propagation evolutions slightly in this case. The larger quintic nonlinearity parameter corresponds to a little more intense central soliton pulse. For high value of soliton order, such as *N* = 2 and 3, which corresponds to high launched power, wide pulse width, weak dispersion, or large cubic nonlinearity coefficient, however, more energy flows to the central soliton pulse which apparently becomes compressed and more intense, as shown in Figs. 1(f)-1(o). Moreover, the soliton pulse exhibits breathing behavior in terms of its intensity and duration. It can be seen that, depending on different soliton order *N* and quintic nonlinearity parameter *R*, the pulse duration and the breathing period are also different. The larger the soliton order *N*, the more narrow the quasi-soliton pulse. In the mean time, for larger value of *N*, more dispersion wave pulses will shed from the Airy wavepacket background and appear beside the central soliton pulse. In comparison, these dispersion wave pulses are weak and generally they will gradually depart far from the central intense pulse. Furthermore, for the case of *N* = 3, more dispersion wave pulses with narrow duration can be observed. In addition, in certain special case, as shown in Fig. 1(m) where *N* = 3, *R* = 0, the soliton pair which consists of the central intense pulse and another weak one beside it will generate. During propagation, the two pulses accompany with each other all the way with their pulse spacing nearly invariant. It is worth mentioning that, in the case of larger values of *N*, the Airy wavepacket only remains a very weak background and even collapses and disappears quickly, which means that the intense nonlinearity, high launched power, wide pulse width, or weak dispersion, is detrimental to the unique propagation property of finite energy Airy pulse.

In order to observe the propagation characteristics of Airy pulse in more detail, the two dimensional temporal evolutions for different values of soliton order *N* and quintic nonlinearity related parameter *R* have also been shown in Fig. 2
. In Fig. 2, *U* = │*u*│/ *u*
_{0}
* _{m}* represents the normalized amplitude, where

*u*

_{0}

*is the maximum of │*

_{m}*u*(0,

*τ*) │. It can be clearly seen that, with increase of propagation distance, the high frequency oscillation side lobes on the leading edge of the Airy pulse will disappear gradually and the leading edge will then become smooth. Only minority of the side lobes are left. Besides, the central intense soliton pulse and the weak dispersion wave pulses sheded from the Airy wavepackets will emerge gradually in the meantime, as just mentioned above. High value of

*N*caused pulse compression and increase of the dispersion wave pulse number can also be seen here again. What is more, Fig. 2 still indicates that, with increase of propagation distance, the temporal wave shape will shift towards the trailing edge of the Airy wavepacket. In fact, this characteristic can also be observed in Fig. 1.

It is worth mentioning that, the Airy wavepackets factually vary with the propagation distance in terms of theirs maximal normalized amplitudes and corresponding temporal positions, as shown in Fig. 3
where *U _{m}* represents the maximum of the normalized amplitude. Obviously, for

*N*= 1, the maximal normalized amplitudes generally tend to increase before decrease and then tend to be nearly fixed finally with only minor fluctuations. In certain range of distance, quintic nonlinearity hardly influences the maximal normalized amplitudes. When the distance is long enough, however, quintic nonlinearity will influence the maximal normalized amplitude extremely. The larger the quintic nonlinearity parameter, the larger the maximal normalized amplitude. Besides, as shown in Fig. 3(d), with increase of distance, the corresponding temporal position of the maximal normalized amplitude linearly shifts towards the trailing and leading edge of the Airy wavepacket alternately. Moreover, in some ranges of distance, quintic nonlinearity hardly influences the temporal position of the maximal normalized amplitude.

In comparison, for *N* = 2 and 3, the things become clearly quite different. With increase of distance, the maximal normalized amplitude exhibits high frequency attenuated oscillation which is around a central value. Depending on different values of *N* and *R*, the oscillation amplitude, oscillation frequency, and the central value, are also different. On the other ha nd, in case of quintic nonlinearity, after experiencing a short distance of weak oscillation, the corresponding temporal position of the maximal normalized amplitude nearly linearly shifts towards the edge of the Airy wavepacket. Concretely, except for the case of *N* = 3, *R* = - 0.03, positive and negative quintic nonlinearity generally makes the temporal position shift towards the trailing and leading edge, respectively. In addition, comparing Fig. 3(e) with Fig. 3(f), one can still note that, larger value of *N* corresponds to larger movement of temporal position. For the case of *R* = 0 where no quintic nonlinearities exist, however, the temporal position tends to a fixed value instead. Accordingly, appropriate quintic nonlinearity can be utilized to manipulate the temporal position of the central soliton pulse sheded from the Airy wavepacket.

Similarly, exploiting the split-step Fourier algorithm, we can also simulate and obtain the evolution property of the Airy pulse in the normal dispersion region. Our preliminary study indicates that the results are quite different since the pulse broadening and optical wave breaking phenomena are supported in this region. The detailed results and discussions will be presented soon in our next work.

## 4. Conclusions

Employing the split-step Fourier algorithm, we numerically simulate the effects of quintic nonlinear parameters and soliton order number on the evolution properties of finite energy Airy pulses in optical fibers. The results show that, in case of large soliton number, quintic nonlinearity will distort Airy pulses dramatically and make them evolve into compressed intense soliton pulses accompanied by several weak dispersion wave pulses. The soliton pulses will vary with the propagation distance in terms of their peak amplitudes and the corresponding temporal positions. With increase of the distance, except for some special cases, positive and negative quintic nonlinearity can respectively make the soliton pulse temporal positions nearly linearly shift to trailing and leading edges of the Airy wavepackets after experiencing their initial decayed oscillations. In case of no quintic nonlinearity, the soliton pulse temporal positions tend to fixed positions. The soliton peak amplitudes also exhibit decayed oscillations but with different oscillation amplitude and central values for different quintic nonlinearity and soliton order number. For small soliton order number, quintic nonlinearity only influences the Airy pulses propagation slightly. On the basis of the interesting research results above, this work enriches the investigations on the nonlinear propagation property of the Airy pulse and soliton generation. it may also be useful for pulse compression. What is more, this work inspires people to exploit appropriate quintic nonlinearity to manipulate the soliton propagation in terms of its temporal position and peak amplitude.

## Acknowledgments

This work is supported by the Key Project of Chinese Ministry of Education (No. 210186), and the Major Project of Natural Science Supported by the Educational Department of Sichuan Province in China (No. 13ZA0081).

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