## Abstract

We study both analytically and numerically the propagation dynamics of an initially chirped Airy pulse in an optical fiber. It is found that the linear propagation of an initially chirped Airy pulse depends considerably on whether the second-order dispersion parameter ${\beta}_{2}$ and chirp $C$ have the same or opposite signs. For ${\beta}_{2}C<0$, the chirped Airy pulse first undergoes an initial compression phase, then reaches a breakup area as depending on the values of $C$, and then experiences a lossy inversion transformation such that it continues to propagate with an opposite acceleration. The chirped Airy pulse is always dispersed during propagation in the case of ${\beta}_{2}C>0$. The impact of truncation coefficient and Kerr nonlinearity on the chirped Airy pulse propagation is also disclosed separately.

© 2015 Optical Society of America

## 1. Introduction

The original Airy wave packet was first found in the context of quantum mechanics by Berry and Balazs more than 30 years ago, as a solution to the Schrödinger equation for a free particle [1]. The Airy wave packet was not experimentally realized in the optical domain until Christodoulides *et al.* made a great breakthrough in 2007. They first introduced the concept of finite-energy Airy beams [2], and then performed the first experiments [3]. Since then, Airy beams, a kind of nonspreading beams that exhibit not only propagation invariance but also transverse acceleration with respect to the propagation direction, have attracted a great deal of interest in recent years [4–6]. The Airy beams not only have the characteristics of quasi-diffraction-free and self-reconstruction but also have an additional interesting feature of transverse acceleration, ie., its trajectory follows a parabolic curve during propagation [2, 3, 7]. Those features have been employed for various applications, such as curved filament generation [8, 9], optical trapping particle [10, 11], high resolution microscopy [12, 13], all-optical routing [14], light bullets generation [15–17].

Recent works introduced further pulsed version of Airy beams to optics by exploiting the analogy between “diffraction in space” and “dispersion in time”. These so-called Airy pulses have attracted growing attention, owing to their remarkable properties of quasi-nondispersive, self-healing and self-bending their dominant intensity peaks, similar to the unique properties of Airy beam. However, there is an important physical difference between spatial and temporal accelerations: although both have the Airy shape, a spatial accelerating beam bends its trajectory in space, whereas only the acceleration of Airy pulse corresponds to a change in the velocity of the intensity peak of the pulse that manifests as self-acceleration or self-deceleration depending on its tails behind or in front of the main peak [18]. In addition, the nonlinear propagation dynamics of spatial Airy beams was investigated and shown some novel behaviors, such as the existence of an additional class of stationary accelerating Airy wave forms [19, 20]. This has stimulated great interest in the problem of Airy pulse propagation from linear to nonlinear regimes [15–17, 21–29]. Under the action of Kerr nonlinearity, soliton will be shed during the process of the Airy pulse propagation [24]. The impact of high-order linear and nonlinear effects on Airy pulse propagation has also been reported [25]. Interestingly, it has been used to experimentally realize the linear light bullets, showing that they are capable of healing the dispersion and nonlinearity induced distortion [15, 16]. Airy pulses are also exploited to not only manipulate the Raman-induced frequency effects [27] but also to control super-continuum generation in highly nonlinear fibers [28]. Soliton pair was generated and engineered through the interaction of Airy pulse [29]. Zhang *et al.* investigated the impact of truncation coefficient on the modulation instability of Airy pulse [30].

The previous works on Airy pulse propagation did not consider the effects of initial frequency chirping. In practice, the laser system based on the chirped pulse amplification technology involves chirp in pulse generation, propagation and amplification. Consequently, the pulses emitted from laser sources are often chirped. Frequency chirp can also be imposed externally and is expected to engineer the laser pulse propagation. It is shown that the frequency chirp can influence laser self-focusing significantly [31–33]. Initial requency chirp have also been used to control supercontinuum generation [34, 35], filamentation [36, 37] and pulse compression [38]. More recently, Milián *et al.* reported the electron density and energy deposition in filament channel can be tuned by changing input chirp of laser pulse [39]. Moreover, we should also consider the effect of initial frequency chirp on Airy pulse propagation. This subject, to the best of our knowledge, remains however unexplored. We are still missing a clear understanding of the role of a chirped phase in Airy pulse propagation. In this paper we are devoted to the study of propagation properties of Airy pulse imposed an initial frequency chirp, and disclosed the novel characteristics.

## 2. Theoretical model

The propagation of optical pulses in an optical fiber can be described by the well-known nonlinear Schrodinger equation (NLSE). To simplify the model and broaden the applicability of the results, we normalize all the variables including the field that is normalized so that its peak input value is unity. The coordinates are normalized as follows: temporal coordinate $T$ is normalized to the incident pulse width${T}_{0}$, propagation distance $Z$ is measured in units of the dispersion length ${L}_{D}={T}_{0}^{2}/\left|{\beta}_{2}\right|$, where ${\beta}_{2}$ is the group velocity dispersion (GVD) parameter. The normalized NLSE then takes the form [40]

## 3. Analytical results of the linear propagation

The linear propagation of Airy pulse is studied by setting $N=0$ in Eq. (1). $U\left(Z,T\right)$ satisfies the following linear partial differential equation:

Equation (2) is readily solved by use of the Fourier-transform method. The general solution of Eq. (2) is give byFor chirped Airy pulse $U\left(T,Z=0\right)=\text{Ai}\left(T\right)\mathrm{exp}\left(aT\right)\mathrm{exp}\left(-iC{T}^{2}\right)$, its Fourier spectrum is given by

Figure 2 shows the positively (left column) and negatively (right column) chirped Airy pulse shapes at different propagation distances by direct calculation according to analytical expression (Eq. (8)) in the anomalous dispersion regime. The linear propagation of positively chirped Airy pulse shown in Fig. 2(a)-2(g) displays an interesting process. It first undergoes slight compression and reaches a focal point; then its Airy pattern breaks down. After passing through such breakdown area, however, the pulses reconstruct a new Airy pattern with opposite acceleration. While it can be clearly seen from Fig. 2(a1)-2(g1) that the negatively chirped Airy pulses are always dispersed with an increasing propagation distance. These features can be understood from Eq. (8). It shows the chirped Airy pulse evolution keeps the airy structure only if $\Theta $ is far from zero. In the anomalous-dispersion regime ($s=1$), the value of $\Theta =1-2CZ$changes from positive to negative for $C>0$as the propagation distance increases while the value of $\Theta $ is always positive for $C<0$. The opposite occurs in the normal-dispersion regime ($s=-1$, $\Theta =1+2CZ$). As a result, chirped Airy pulses can switch direction between acceleration and deceleration for $sC>0$, while it always dispersed for $sC<0$. Therefore, the linear propagation dynamics of chirped Airy pulse depends strongly on the signs of dispersion and chirp. It can switch between acceleration and deceleration for $sC>0$, while it always dispersed for $sC<0$.

## 3. Numerical results

To confirm the propagation dynamics of chirped Airy pulse obtained from the analytical analysis, we model Airy pulse propagation with the Eq. (1) in an optical fiber by using the well-known split-step Fourier method [42]. To get a better visualization of the low-intensity parts of the Airy pulse propagation, hereafter, we plot the evolution of temporal and spectral absolute amplitude $\left|U\left(Z,T\right)\right|$ and $\left|\tilde{U}\left(Z,\omega \right)\right|$ and instead of the corresponding intensity ${\left|U\right|}^{2}$and ${\left|\tilde{U}\right|}^{2}$as a function of propagation distance. For $N=0$, Eq. (1) is only capable of modeling the effects of GVD on optical pulses propagating in a linear dispersive medium. Figure 3(a) demonstrates the linear propagation of an initially unchirped Airy pulse, showing that it maintains its all remarkable properties of the ideal self-decelerating Airy pulses over an finite propagation distance. The shape of self-decelerating unchirped Airy pulse remains quasi-invariant over several dispersion lengths while again the intensity features tend to “freely decelerate” until dispersion dominates. It is consistent with results in Ref [2]. However, this behaviors change drastically if the Airy pulse has an initial frequency chirp. Figures 3(b) and 3(c) display the dynamics of the Airy pulse with positive and negative frequency chirp in the case of anomalous dispersion regime respectively. Clearly, the linear dynamics of chirped Airy pulses are quite different when compared to that associated with unchirped Airy pulse, as depicted in Fig. 3(a). It is evident that the sign of chirp parameter $C$ plays a critical role.

A comparison of Figs. 3(b) and 3(c) shows that, the effect of frequency chirp on the propagation of Airy pulse depends on whether $C$ is positive or negative. When the incident Airy pulse was positively chirped ($C>0$), the propagation of Airy pulse shown in Fig. 3(b) can be separated to three regimes of interest: it undergoes compression in the first 2.2 propagation distance; then the pulse pattern breaks up with further increasing of the propagation distance; Finally, a new Airy pattern with the rotational symmetry distribution with respect to the input pulse was regenerated. The amplitude of new Airy pulse is smaller than that of original Airy pulse. However when the incident Airy pulse was imposed a negative chirp ($C<0$), the Airy pulse was always dispersed during the propagation. Their lobes experience an apparent acceleration, exactly opposite to its original deceleration. The numerical simulations confirm the novel propagation dynamics of chirped Airy pulses predicted by analytical analysis. In addition, Fig. 4 displays a comparison of pulse shapes between analytical and numerical results at some representative propagation distances. All solid redlines overlap with red dash lines. Once again, we get very good qualitative agreement between analytical results and numerical simulation of the NLSE. Although, this process is similar to that reported in [21], where the reason is the dominant contribution of third-order dispersion parameter, the physical reason behind such propagation dynamic is different.

To have a better understanding of the propagation dynamics of chirped Airy pulse, let us look into the corresponding spectra. The spectral evolution of chirped Airy pulses have dramatic difference compared to that of unchirped Airy pulse as shown in the right column of Fig. 3. The spectrum of unchirped Airy pulse shown in Fig. 3(d) always keeps the Gaussian distribution without any change along the propagation direction. While the spectral evolution of chirped Airy pulse has oscillatory tail behind or in front of its main peak, depending on whether chirp ($C$) is negative or positive. This agrees well with analytical results (Fig. 1). Furthermore, its shape is also not changed during propagation. We also performed the linear propagation of a chirped Airy pulse in a fiber with normal dispersion (data not shown), indicating that the opposite occurs. The features of chirped Airy pulse propagation seen in Fig. 3 can be understood qualitatively as follows. The dispersion-induced chirp is positive or negative, depending on whether dispersion is normal or anomalous. When ${\beta}_{2}C>0$, the dispersion-induced chirp adds to initial input chirp because the two contributions have the same sign. In the case of ${\beta}_{2}C<0$, the situation changes dramatically, the contribution of the dispersion-induced chirp is of a kind opposite to that of the initial input chirp. Naturally, the same propagation dynamic occurs, if the incident chirped Airy pulses are launched with a reversed acceleration.

Figure 5 shows the temporal evolution of chirped Airy pulse as a function of propagation distance for different values of $C$ ranging from 0.1 to 2.0. It illustrates how much an initial chirp can modify the propagation behavior of Airy pulse. For $C=0.1$, the chirped Airy pulse just experiences an initial compression, and then breaks up in the 20 propagation distances. There is no new Airy pattern formation. With the increasing the chirp, new Airy pattern with reversal acceleration can be clearly seen. The propagation distance required for new Airy formation decreases with increasing chirp. It can be clearly seen from Fig. 6(a) that the size of breakup area (${L}_{B}$, measured from one amplitude maximum to another and marked in Fig. 4(a1)) becomes smaller and smaller with an increasing $C$. For a small chirp ($C\le 0.1$), ${L}_{B}$ becomes very large such that the size diverges. In addition, the new formed Airy pulse carries more energy for larger chirp. But the Airy pulse is also distorted in shape.

When the self-accelerating chirped Airy pulses have a large value of truncation coefficient, their unique features disappear rapidly. Their unique features therefore strongly depend on the value of truncation coefficient. Figure 7 shows temporal evolutions of chirped Airy pulse with different values of truncation coefficient over 2 dispersion lengths. The process of compression, breakup and regeneration are almost unchanged for different truncation coefficients. However, the compression factor, defined as the ratio of maximum peak intensity to input peak intensity, decreases with increasing truncation coefficients. The size of breakup area slightly increases as the truncation coefficient is increased. The rate of the size growth depends on the initial chirp [Fig. 6(b)]. The Airy pulses with larger truncation coefficient will loss more energy through the breakup area.

Under the action of Kerr nonlinearity, the Airy pulse was distorted in the form of soliton shedding and dispersion background during propagation [24]. Figures 8(b)-8(f) show the temporal evolution of Airy pulse for different Kerr nonlinearity. Compared the case of linear propagation shown in Fig. 8(a), the compression ratio is reduced owing to the presence of nonlinear self-focusing as nonlinear parameter $N$ is increased. For $N\le 1$, the nonlinear self-focusing effects play an increasingly more important with increasing $N$, then become comparable to the dispersion effects. The combined effects of GVD and Kerr nonlinearity are to reduce the compression ratio; the size of breakup area widens but, at the same time, the resulting energy loss increased. The situation changes dramatically for $N>1$. In this case, the nonlinear self-focusing effect dominates the dispersion effect, which completely disrupts the phase evolution of chirped Airy pulse. Under the action of an even stronger nonlinearity, as shown in Figs. 8(g)-8(i), the high energy Airy pulse enters a soliton shedding regime, similar to that reported in [24]. The trajectories of the shedding solitons are affected by successive collisions with the side lobes and dispersion background. The number of shedding soliton increases with increasing $N$. As a consequence, the rotational symmetry of propagation process has been changed dramatically.

## 4. Conclusion

In summary, we have investigated the propagation dynamics of truncated Airy pulse with an initial frequency chirp imposed in optical fibers by means of direct simulation and theoretical analysis. The analytical expression for chirped Airy pulse propagation is obtained. We find that, when the second-order dispersion parameter ${\beta}_{2}$ and chirp $C$ have the opposite signs, the chirped Airy pulse reaches the breakup area after experiencing an initial compression, then undergoes a lossy inversion, and finally continues to travel with an opposite acceleration. The size of breakup area, measured from one amplitude maximum to another, decreases with increasing $C$ or decreasing truncation coefficients. While the chirped Airy pulses are always dispersive in the case of ${\beta}_{2}C>0$. The analytical results are very well agreement with the numerical simulations. In the weakly nonlinear propagation regime, the compression ratio is reduced that accompanies severe energy loss as the nonlinear parameter $N$ is increased ranging from 0 to 1. With further increasing Kerr nonlinearity ($N>1$), such stronger Kerr nonlinearity brings the chirped Airy pulse propagation into the regime of multiple soliton generation and successive collisions.

## Acknowledgments

The authors thank the referees for enlightening comments on the analytical results of chirped Airy pulse propagation. This work was supported by the Program of Fundamental Research of Shenzhen Science and Technology Plan (Grant Nos. JCYJ20140828163634005, JCYJ20140418095735599), the Natural Science Foundation of SZU (Grant Nos. 201449, 201450) and the Hunan Provincial Natural Science Foundation of China (Grant Nos. 13JJ4108, 15JJ2036).

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