1. Introduction

The finite-energy Airy beams, which were predicted theoretically and investigated experimentally by Siviloglou et al. in 2007 in the context of optics [1, 2], have attracted much attention [3–15] due to their unusual properties including self-acceleration [2], quasi-diffraction-free [3] and self-healing [4]. These properties have been employed for various applications, including optical trapping particle [5–7], curved filament generation [8, 9], optical routing [10] and vacuum electron acceleration [11, 12].

The temporal Airy pulses, known as the pulsed version of the Airy wave packets, which were also studied [16–26], leading to a broader understanding of the propagation of the Airy pulses from linear to nonlinear regimes. In the linear regime, the evolution of Airy pulses launched into a fiber under the dominant negative and positive third-order dispersion has been reported [17,18]. Chong et al. experimentally investigated a versatile linear Airy-Bessel light bullet by containing Airy distribution in time domain with Bessel distribution in spatial domain [19]. Abdollahpour et al. studied the spatiotemporal airy light bullets [20]. In the nonlinear regime, the influences of instantaneous Kerr nonlinearity and the Raman scattering on the propagation of Airy pulses have been studied widely [21–26]. In the instantaneous Kerr media, it is found that a sech-type soliton was shed from the Airy pulse in a single mode fiber in the presence of self-phase modulation and anomalous dispersion, and the remaining Airy pulse still continues to exhibit the unique property of acceleration in time [21].

Let us revisit the physical mechanism of a traditional soliton. It is well-known that the chirp contributions induced by the self-phase-modulation (SPM) and anomalous group velocity dispersion (GVD) cancel each other during the pulse propagation, and a chirp-free pulse with unchangeable profile is observed when the cancelation is complete [27]. For the Airy pulse, on the other hand, the linearity makes its main lobe accelerate (with an increasing group velocity) or decelerate (with a decreasing group velocity) during propagation, which can be regarded as a constant “gravity” on the pulse [3,16]. A natural question is raised that, is there any nonlinearity that can counteract such acceleration (deceleration) of the accelerating (decelerating) Airy pulse?

Very recently, we have found that the highly noninstantaneous Kerr nonlinearity imposes a approximately constant “force” on the “mass center” of any optical pulse with finite energy during propagation [28]. Therefore, such highly noninstantaneous Kerr nonlinearity might be the candidate to counteract and even balance the linear “gravity” of the decelerating Airy pulse. In this paper, we investigate the propagation of the decelerating Airy pulse in the highly noninstantaneous Kerr media. It is found that the highly noninstantaneous nonlinearity does counteract the linear “gravity” of the Airy pulse as expected to change its deceleration, which paves the way to the dynamic control of an Airy pulse in the Kerr media. The linear “gravity” of the decelerating Airy pulse can be balanced by choosing a specific power of the pulse, producing an unchangeable (neither decelerating nor accelerating) Airy-type solitary wave within several dispersion lengths.

2. Theoretical model and numerical results

We consider the propagation of an optical pulse in the anomalous dispersion regime in highly noninstantaneous Kerr media, which can be described by the dimensionless temporally nonlocally nonlinear Schrödinger equation [28, 29]

The initial self-decelerating Airy pulse can be written as: $u(0,t)=\sqrt{P\mu (\alpha )}Ai(-t)\text{exp}(-\alpha t)$, where α is the decay parameter, $P={\int }_{-\infty }^{\infty }{\left|u\right|}^{2}dt$ denotes the total power, and μ(α) is a normalized coefficient.

To see how the nonlinearity and nonlocality impacts on the decelerating Airy pulse, we compare the nonlinear refractive index changes and the frequency chirp induced by the SPM of a Gaussian pulse in instantaneous media and a decelerating Airy pulse in highly noninstantaneous Kerr media. The refractive index changes and the frequency chirp are given by $\mathrm{\Delta }n={\int }_{-\infty }^{\infty }R(t-\tau )\left|u(0,\tau )\right|d\tau$ and $\delta \omega (t)=-\delta z\partial \left(R(t)\otimes {\left|u(t)\right|}^2\right)/\partial t$, with δz denotes a very short propagation length [27]. For the instantaneous case [see Fig. 1(a)], it is well known that the pulse and the nonlinear refractive index change simultaneously reach their maxima. The chirp induced by the SPM is negative near the leading edge and becomes positive near the trailing edge, and is nearly linear across the central region of the pulse. Note that the chirp induced by anomalous GVD is positive near the leading edge and negative near the trailing edge. Therefore, these two chirp contributions may balance each other during the pulse propagation, leading to the unchangeable profile of the pulse, i.e., soliton [27]. For the highly noninstantaneous case [see Fig. 1(b)], it can be easily seen that the nonlinear refractive index change is delayed to the pulse and raised relaxedly, which means that the nonlinear refractive index change induced by the main lobe affect the minor lobes of the decelerating Airy pulse. It is interesting that the chirp of the initial decelerating Airy pulse induced by SPM (nonlinearity) looks like the “inverted image” of the pulse profile [28], which is always negative over the whole region of the pulse. Such negative chirp equivalently imposes a constant “force” on the decelerating Airy pulse in the positive direction. Note that the linearity here can be regarded as a constant “gravity”, of which the direction is opposite to the nonlinear “force”. Therefore, the dynamics of the decelerating Airy pulse depends on the competing cooperation of these two “forces”.

 

Fig. 1 (a) The distributions of the intensity, Δn and δω(t) of a Gaussian pulse in instantaneous media. (b) The distributions of the intensity, Δn and δω(t) of a decelerating Airy pulse in highly noninstantaneous Kerr media with α = 0.02 and T = 100.

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Next, we consider the the evolution of the decelerating Airy pulse in highly noninstantaneous Kerr media by simulating Eq. (1) in terms of the split-step-Fourier method. Figures 2(a)–2(c) display the temporal evolution the decelerating Airy pulse with T = 100 for different power in highly noninstantaneous Kerr media. For comparison, the linear case (P = 0) is also presented in Fig. 2(a), which exhibits a well-known trajectory of t = −z²/4 within several dispersion length. For the case of a weak power P = 200, it is shown by Fig. 2(b) that the nonlinearity restrain the deceleration of the decelerating Airy pulse. The temporal evolution of Airy pulse as a function of propagation distance with the initial power P = 1500 is shown in Fig. 2(c). Due to the strong nonlinear effect, the trajectory of the decelerating Airy pulse accelerates to the opposite site and the energy of the main lobe of decelerating Airy pulse decays.

 

Fig. 2 (a)–(c) represent the temporal evolution of the decelerating Airy pulse with T = 100 as a function of propagation distance for different power in highly noninstantaneous Kerr media: (a) P = 0, (b) P = 200, and (c) P = 1500. (d) The trajectories of the main lobe of the decelerating Airy pulse with different P.

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In order to see the influence of the initial power more clearly, we also calculate the evolution of the “mass center” of the main lobe in the form of $〈T_c〉={\int }_m^{n}t{\left|u(t,z)\right|}^{2}dt/{\int }_m^n{\left|u(t,z)\right|}^{2}dt$ with m and n denoting the upper and lower limits of the main lobe of decelerating Airy pulse, which is shown in Fig. 2(d). It is clearly shown that the trajectory of the 〈T_c〉 depends on the power of the pulse: the noninstantaneous nonlinearity restrains and even accelerates the decelerating Airy pulse. This interesting phenomenon provides a method to control the dynamics of the Airy pulse in the nonlinear regime.

The investigations above indicate that the trajectory of the main lobe of the decelerating Airy pulse can be continuously manipulated by adjusting the power of the pulse. In the following, we would like to discuss physically when the noninstantaneous nonlinearity can balance the linear “gravity” of the decelerating Airy pulse. It is well known that the linear “gravity” of the decelerating Airy pulse is |F₁| = 1/2 [3], while the highly noninstantaneous Kerr nonlinearity imposes a approximately constant “force” on the “mass center” of any optical pulse with finite-energy is given by [28]

The temporal evolution of the decelerating Airy pulse with the balanced power P_b = 703.81 in highly noninstantaneous Kerr media is shown in Fig. 3(a). The inset of Fig. 3(a) displays the trajectory of the main lobe, which shows the evolution more clearly. It can be easily seen that the pulse profile and the trajectory of the decelerating Airy pulse are nearly unchangeable in several dispersion lengths, i.e. the decelerating Airy pulse changes to the Airy-type solitary wave. The intensity distribution of the decelerating Airy pulse in highly noninstantaneous Kerr media at different propagation distances is shown in Fig. 3(b). It is interesting to find that the wave profile keeps in Airy-type, and the Airy pulse neither decelerates nor accelerates during propagation. But the energy of the main lobe flows to the minor lobes due to the noninstantaneous effect, which can be seen clearly between the pulse profiles z = 0 and z = 3. Because the energy of the main lobe decreases, the balance between the linear “gravity” and the nonlinear “force” is destroyed, the main lobe of the Airy pulse decelerates with increasing the propagation distance, as shown in Fig. 3(c).

 

Fig. 3 (a) The temporal evolution of the decelerating Airy pulse as a function of propagation distance in highly noninstantaneous Kerr media with the balanced power within several dispersion lengths. Inset: the trajectory of the main lobe of the decelerating Airy pulse. (b) The intensity distribution of decelerating Airy pulse in highly noninstantaneous Kerr media at different propagation distances. (c) The temporal evolution of the decelerating Airy pulse as a function of propagation distance in highly noninstantaneous Kerr media with the balanced power within z = 3 to z = 6.

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Generally, the Airy-type solitary wave is resulting from a new method of controlling Airy pulse in highly noninstantaneous Kerr media. From Eq. (2), it is worth mentioning that, the nonlinear “force” of the pulse in highly noninstantaneous Kerr media is proportional to the ratio P/T [28], so the trajectory of the Airy pulse can be controlled by the initial power. When the power is suitable, the linear “gravity” is equal to the nonlinear “force” of the Airy pulse in highly noninstantaneous Kerr media, and the Airy-type solitary wave can be obtained. It should also be pointed out that the balanced power P_b is influenced by the truncation coefficient α, which can be seen in Fig. 4(a). Figure 4(b) shows the trajectories of the main lobe of the Airy-type solitary waves with different truncation coefficients. It is significant to find that, the smaller the truncation coefficient is, the more easily the decelerating Airy pulse changes to the Airy-type solitary wave.

 

Fig. 4 (a) The balanced power of decelerating Airy pulse as a function of the truncation coefficient in the highly noninstantaneous Kerr media. (b) The trajectories of the main lobe of the Airy-type solitary waves with different truncation coefficients.

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The discussion for the Airy-type solitary wave is based on Eq. (1), where the losses are absent. However, the effect of losses is always unnegligible for the reorientational nonlinearity. Qualitatively, the losses destroy the counterbalance between the linear and nonlinear “forces”: the nonlinear “force” decreases with the decay of the pulse power while the linear “force” still keeps constant. As a result, the pulse begins to decelerate with an increasing deceleration during propagation. Practically, the losses can be avoided as far as possible by specifically choosing the medium and the wavelength of the pulse. Take CS₂-filled photonic crystal fibers as an example, the scale of the dispersion length under the highly noninstantaneous condition is in the order of one meter [28], and the losses can be neglected in such a distance by choosing the wavelength within the transparent window of the CS₂ liquid (from 0.5 to 3μm) [29].

3. Conclusion

In conclusion, we have investigated the dynamics of a decelerating Airy pulse in the highly noninstantaneous Kerr media. It is found that the highly noninstantaneous nonlinearity does counteract the linear “gravity” of the decelerating Airy pulse as expected to change its deceleration, which paves the way to the dynamic control of an Airy pulse in the Kerr media. There is a competition between the two opposite “force” imposed on the decelerating Airy pulse by the linearity and nonlinearity, and the competition can be balanced by choosing a specific power of the Airy pulse, producing an unchangeable Airy-type solitary wave within several dispersion lengths. The dynamics of a decelerating Airy pulse in the highly noninstantaneous Kerr media could be controlled, creating new opportunities for optical steering and manipulation.