## Abstract

We propose theoretically various kinds of filaments via the Mathieu modulation. Our results indicate curved filaments, in-phase and out-of-phase quasi-solitons and nonlinear light bullets can be formed successfully in air. Through calculated initial Mathieu accelerating beam (MAB), curved filament is able to propagate along a predesigned elliptical trajectory. By transforming the MAB into an axial symmetrical structure with in-phase and out-of-phase modulations, we obtain two kinds of quasi-solitons in air, respectively. The latter case can even propagate in a breathing fashion. With a ring structure of MAB, we successfully form a light bullet in air that generates a chain of intensity peaks over extended distances. These unique filaments can offer significant advantages for numerous applications, such as micro engineering of materials, THz radiation generation and attosecond physics.

© 2017 Optical Society of America

## 1. Introduction

Filamentation has stimulated growing research interest since the first observation of filament generated by an intense femtosecond laser pulse in air [1]. Owing to a number of nonlinearities during the propagation of Gaussian beams in air [2–5], the counterbalance between the optical Kerr nonlinearity and plasma de-focusing effect leads the beam to become a self-guided propagating mode and numerous peculiar phenomena occur, including spectral broadening, self-pulse compression and plasma channel, etc. Possible applications of filaments have also been proposed and demonstrated in laser induced electrical discharge [6,7] particle acceleration [8], LIDAR remote sensing [9], THz radiation generation [10], and pulse compression [11].

The continuing quest on how to manipulate filaments propagating in an expected way has constantly been pursued in the intense femtosecond laser pulse area. Recently, research efforts have been devoted to generate unique filaments via utilizing the nondiffracting and self-healing properties of Airy beams [12–15]. In 2009, Pavel Polynkin *et al* proposed a scheme that uses ultraintense Airy beams to generate curved plasma channel in air for the first time [16]. This unusual propagation regime inspires many ways of controlling filaments so that multiple applications can be achieved. One instance is that Matteo Clerici *et al* guided the electric discharge around objects in air in a controllable manner by taking advantage of the nondiffracting property of Bessel and Airy beams [17]. Furthermore, in the way of adding an Airy profile in time domain, Daryoush Abdollahpour *et al* illustrated intense Airy-Airy-Airy (Airy^{3}) light bullets [18], showing that the Airy^{3} light bullets possess the self-healing nature in both temporal and spatial domains. The self-reconstruction characteristic exhibited by Airy beams contributes to form an intense steep intensity peak on axis on the condition of using ring-Airy wave packets. Unfortunately, the Airy beams always propagate along a parabolic trajectory that always deteriorated by nonlinearities and cannot be predesigned. As such, when filaments are generated by Airy beams, Kerr and multiphoton nonlinearities would make the main lobe with high powers are reshaped into a multifilamentary pattern [19].

During recent years, on the one hand, numerous efforts have been dedicated to moving the main lobe’s intensity of Airy beams along a predesigned trajectory [20–23]. The Quasi-Airy beams proposed by Yixian Qian *et al* serve as a typical example [24]; on the other hand, novel nonparaxial accelerating beams have been introduced theoretically and demonstrated experimentally [25,26]. Those beams are not subject to a single parabolic trajectory. Lately, Peng Zhang *et al* found that the circular nonparaxial accelerating beams are only one special case of the Mathieu accelerating beams (MABs) [27]. In this concept, the MAB can propagate in a different fashion travelling along an ellipse with any predesigned ellipticity. The wave packet modulated by radial and angular Mathieu functions results in a scalable acceleration control without the paraxial limit. In our previous work, we have found the amplitude modulation of a Mathieu beam offers a new approach to control filaments’ onset, distribution and elongate the whole length of filamentation [28]. From the results simulated by Vincotte et al, the stability of femtosecond optical vortices propagating in the atmosphere was verified for the first time [29]. However, the value of Mathieu modulation has never been fully disclosed in the filamentation regime in this paper, because there are abundant kinds of wave packets initialed by certain modulations. For instance, concerning spatial solitons formed with the Airy beams in nonlinear media, in-phase and out-of-phase initial packets are two kinds of ways to form novel breathing solitons [30]. If rotating the Airy modulation into a rotational symmetry structure, a ring Airy wave packet can also be formed. Naturally, this brings about following relevant questions: Whether could an MAB with high intensity generate a filament propagating along a predesigned trajectory when assisted by nonlinearities? If so, would the wave packet with Mathieu modulations, including in-phase and out-of-phase situations as well as ring structure, generate novel phenomena as depicted in previous works?

In this context, we theoretically investigate various kinds of filaments via the Mathieu modulation. We find that curved filament could propagate along a predesigned elliptical trajectory. By transforming the MAB into an axial symmetrical structure with in-phase and out-of-phase modulation, different kinds of quasi-solitons in air are constructed. Finally, we successfully form a light bullet in air using intense ring MAB that generate a chain of intensity peaks. For their potential applications, the elliptical trajectory of MAB and two kinds of quasi-solitons may produce different microchannels; the bullet may be beneficial to high aspect ratio nanochannel machining, supercontinuum and THz generation.

## 2. Model

In the nonlinear optical regime, several types of well-established numerical models such as Forward Maxwell’s Equation (FME), Nonlinear Envelop Equation (NEE) and Non-linear Schrödinger Equation (NLSE) were used in the simulations. These propagation equations are under one roof that is named Unidirectional Pulse Propagation Equation (UPPE) [31–33]. Even though UPPE put a much heavier burden on the computer when compared to other envelope models with paraxial approximation, it is necessary to concern the non-paraxial nature of MABs. The propagation equation along the propagation direction z for the pulse envelop $E\left(x,y,z,t\right)$in the Fourier domain reads:

The nonlinear polarization${P}_{\omega ,{k}_{\perp}}$caused by the Kerr effect possesses the relation with the pulse analytic signal as:

where${n}_{2}$and${n}_{0}$are the Kerr nonlinearity coefficient and linear refractive index, respectively. Here, the instantaneous Kerr effect parameter${n}_{2}=0.96\times {10}^{-19}{\text{cm}}^{\text{2}}/\text{W}$is determined by the critical power for self-focusing ${P}_{cr}=3.77{\lambda}_{0}^{2}/8\pi {n}_{2}$equivalent to 10GW in air. The new time variable$\tau =t-k{\text{'}}_{0}z$is utilized changing the reference frame from laboratory to local time with the group velocity${v}_{g}=k{\text{'}}_{0}{}^{-1}$.The nonlinear absorption and plasma defocusing are described by a current density spectrum${J}_{\omega ,{k}_{\perp}}$corresponding to the optical field ionization of the air with oxygen molecules density${\rho}_{nt}=5\times {10}^{18}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\text{cm}}^{-3}$.

The electron density generated by MPI and avalanche is modeled by the relation:

It has to be emphasized that the third-harmonic generation is discarded in Eq. (3) for the nonlinear polarization$P$. The reason behind it is that we intend to investigate the evolution process of filamentation instead of third-harmonic generation. To be specific, in the work explored by N. Aközbek et al, the energy conversion efficiency from fundamental to third-harmonic is never up to 0.4% in air, and when the input power is above${P}_{cr}$, the conversion efficiency does not increase with further increasing input power [34]. Therefore, for the filamentation in air, the intensity of third harmonic would not cause significant influence on the refractive index, which is not similar to the filamentation in noble gases.

MAB is one of the solutions for Helmholtz equation in the elliptical coordinate system [35]. Assuming that$\left(y,z\right)=\left(h\mathrm{sinh}\xi \mathrm{sinh}\eta ,h\mathrm{cosh}\xi \mathrm{cosh}\eta \right)$is the transformation from elliptical coordinate system to Cartesian coordinate system, where the interfocal separation$h={\left|{a}^{2}-{b}^{2}\right|}^{1/2}$is determined by the two semiaxes$a,b$of the ellipse [Fig. 1(a)], and$\eta \in [0,\text{\hspace{0.05em}}\text{\hspace{0.05em}}2\pi )$, $\xi \in [0,\text{\hspace{0.05em}}\text{\hspace{0.05em}}\infty )$are the angular and radial variables, respectively. In the elliptic cylindrical coordinate system, the MAB can be described as$M\left(\xi ,\text{\hspace{0.05em}}\text{\hspace{0.05em}}\eta \right)\text{=}R\left(\xi \right)\Theta \left(\eta \right)$, where $R\left(\xi \right)$and$\Theta \left(\eta \right)$are the solutions of modified and canonical Mathieu differential equations [Eq. (6)] that are split from Helmholtz equation.

Here, $\alpha $is the parameter to confine the energy to be finite, $H\left(y\right)$is a Heaviside function, Re represents the real part. Note again that the modulation just takes place in the$y$direction approximately for$y>0$due to the Heaviside function. For simplicity, we can add a Gaussian modulation in the$x$direction so that the accelerating actions can be observed in the$y$direction. Furthermore, we can make this special modulation axial symmetry or even rotational symmetry, which construct not only the in-phase and out-of-phase initial packets, but also the ring Mathieu bullet. In our work, we adopt the parameter b as 30 cm and select$a=4,\text{\hspace{0.05em}}\text{\hspace{0.05em}}2,\text{\hspace{0.05em}}\text{\hspace{0.05em}}1\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{mm}$to achieve different trajectories in the y direction whose corresponding initial normalized amplitude in the y direction at $z=0$ is sketched in Fig. 1(b).

For the initial femtosecond laser pulse, the temporal duration of a Gaussian profile is ${t}_{\text{FWHM}}=\sqrt{2\mathrm{ln}2}{t}_{p}$ with${t}_{p}=42.5\text{fs}$ and central wavelength is$800\text{nm}$.

## 3. Discussion

Let us start with the Mathieu accelerating beams which possess the corresponding predesigned ellipticities and those beams’ initial amplitude profiles are sketched in Fig .1(b), respectively. In order to examine the beam propagation dynamics, Eqs. (1) and (3) are integrated numerically by using initially pulses taken with norm Gaussian in the x direction as:

Figure 2 illustrates 2D representations of the filament intensity for three cases in transverse plane $(x=0,y)$ as a function of propagation distance$z$. Apparently, all of them propagate along the elliptical trajectories as predesigned by the two semiaxes$a,b$. Since the main lobe having a larger size than satellite lobes carries the highest power, it will be clamped after several centimeters of propagation and the corresponding clamping intensities for three cases can be seen intuitively in Fig. 3(b). By contrast, it is clear that the length of curved filament tends to be shorter with a higher ellipticity. As one can notice, unlike the beam propagating in the linear regime, the intensity of the main lobe is constantly clamped into a relatively stable level exhibiting one or several plateau until the filament terminates. During this evolution process, the beam patterns of the main lobe are not only influenced by two nonlinearities, Kerr and plasma defocusing effects, but also significantly susceptible to the initial phase distribution. On the one hand, according to the energy replenishment theory of filamentation [38], the narrow core of the filament is refilled by the surrounding laser energy that is named energy reservoir. Consequently, if the background energy is insufficient to support central area, the filament will end up immediately; on the other hand, when adjacent lobes are introduced a$\pi $phase lag, the filaments will form independently within each lobe possessing enough power [39]. Accordingly, for MABs with a$\pi $phase shift between adjacent lobes, the lobes would propagate separately at the beginning of propagation and then numerous filaments are generated, as shown in Fig. 2, and the length of filament decreases with higher order lobe due to different powers. Therefore, the main lobe carrying the highest power determines the length of the curved filament. In particular, with higher ellipticity the number of filaments decreases while more satellite lobes are involved just for the sake of smaller size leading to lower power. The power of the main lobe from Case (a) to Case (c) is$1.25{P}_{cr}$, $1.50{P}_{cr}$and$1.75{P}_{cr}$, respectively. This phenomenon can also be explained successfully by the energy replenishment theory of a single filament. As addressed above, it implies that the curved filament with$a>b$as demonstrated in Ref [27]. in the linear regime cannot be generated due to the extremely small sizes of lobes which are merely several micrometers, namely insufficient background energy for each single lobe to accomplish filamentation.

According the pioneer works presented by Bérge and Brandon G. Bale et al. [40, 41], a dipolar structure with opposite phase will never fuse. However, it is necessary to point out in this work that it involves a more complex process during filamentation process. As illustrated before, different lobes generate filaments with different lengths. The interaction among nonlinearities would change the phase distribution of the beam when compared with the case propagating in the linear regime. Consequently, at the post stage of filamentation, the phase difference between adjacent lobes is not exactly$\pi $. When the interacting light beamlets propagate in Kerr-type media, two filaments separated within a certain distance would amalgamate in the case of no exact$\pi $phase difference. Therefore, at the end of filamentation, the energy diffracted by linear effects can overlap resulting in a blurry distinction between different lobes.

To gain further details of this interesting behavior, we monitor in Fig. 3 the displacements of the main lobes and peak intensities for three cases as a function of propagation distance. The left column of Fig. 3 shows a comparison of the elliptical trajectories with three different ellipticities during the nonlinear propagation. At the propagation distance$z=20\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{cm}$, the displacements of them are$1.5\text{mm,}\text{\hspace{0.05em}}\text{\hspace{0.05em}}0.8\text{mm,}\text{and}0.55\text{mm}$, respectively. It is worth mentioning that the filament of case 1 terminates at$z=20\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{cm}$and it is shorter than other two cases. Therefore, the limited length of filament with high ellipticity is the key factor to confine the higher level of displacement. As we can see in Fig. 3(b), the intensities of three cases are all clamped at about$70\text{\hspace{0.05em}}\text{TW}/{\text{cm}}^{\text{2}}$with a slight difference [42]. For case 1 with the smallest main lobe, the intensity increases gradually along propagation direction and it forms a single intensity plateau at$z=14\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{cm}$, indicating that the filament is almost dominated by the Kerr nonlinearity until it is clamped at$z=14\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{cm}$. After this distance, the filament is so thin that Kerr effect is not capable of focusing the energy defocused by plasma and diffraction and thus the intensity would decrease gradually. For case 3 with the largest main lobe, it is obvious two intense peaks are formed at$z=10\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{cm}$and$z=22\text{\hspace{0.05em}}\text{cm}$, respectively. This phenomenon exhibits distinct difference compared with the intensity distribution along the propagation direction in the linear regime. Before the distance$z=10\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{cm}$, Kerr effect dominates the evolution of energy flux so that intensity grows up. Nevertheless, when plasma density is high enough to overcome the Kerr nonlinearity, the intensity would decrease between the distances$z=10\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{cm}$and$z=14\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{cm}$. These results give the evidence that the nonlinearities dominate the propagation dynamics of the single main lobe before the filaments terminate while the trajectories are governed by the linear accelerating nature of MABs.

Even though the MABs exhibit a different kind of trajectory when compared with the parabolic Airy beams [43–46], they share plenty of similarities in weakly nonlinear regime. When the initial size of main lobe is small, the main lobe of the beams accelerates transversely faster, shown in Fig. 3(a). The intensity profile does show plateaus because the self-defocusing prevents intensity to exceed a certain value predicted by estimations [47]. At the beginning of propagation, the individual lobe shrinking is also observed due to the Kerr nonlinearity. The filaments at a certain propagation distance would split up and the split-off ones do not exist far and die out in disappearance of the accelerating beam support. As the same family of accelerating beam, these phenomena suggest that the nonlinear dynamics of MABs are governed by the same principles applying to Airy beams.

Next, we carry out the analysis by using the energy flow and Poynting vector to illustrate the field configurations of case 2 at different propagation distances. In Fig. 4, we can see clearly the direction and the strength of energy flow that are demonstrated intuitively during the process of MABs propagation in nonlinear regime. At the beginning propagation stage$z=10\text{cm}$, as shown in Fig. 4(a), the direction of all lobes’ energy flow is partially towards the center and partially towards the positive direction of y-axes. There is no doubt that the direction of energy toward positive y direction is caused by the accelerating nature of MABs and the direction toward the center is due to the Kerr nonlinearity. Upon further propagation, the Kerr self-focusing effect will first disappear in the satellite lobes and this part of energy evolves toward outside, as shown in Fig. 4(b). Note that the direction of the main lobe’s energy flux almost keeps the same trend as that of the former stage but is slightly weakened in the direction toward center. This is because the filaments of satellite lobes are ended up, leading the diffraction to dominate the evolution. In regarding to the main lobe, the size in the y direction becomes more compressed but the acceleration is higher than that at$z=10\text{cm}$, as shown in Fig. 2(b). At the same time, the secondary lobe is closer to the main one, which causes higher field interference between each other. Thus, it is inevitable that the acceleration of the main lobe is higher, namely, the energy flux in the y direction is strengthened [arrows in Fig. 4(b)]. When the filament of the main lobe begins to terminate, the movement of energy flux is illustrated in Fig. 4(c). By contrast, the gaps between adjacent lobes become vague in this case and hence the lobes are not distinguishable. However, it is readily seen that accelerating speed in the y direction is not weakened even though filaments disappear since the intensity in this case is relatively low to avoid being dramatically affected by nonlinearities.

The discussions above provide us with a crucial fact that MABs can propagate in a controllable fashion in the intense femtosecond laser pulse area. This unique property can be used as a prerequisite to produce numerous interesting phenomena in the nonlinear regime that are useful for their potential applications. One hot concern is that using accelerating beam in a special nonlinear media to generate spatial solitons. These media are somehow uncommon in our nature, which may prevent their potential applications. Here, we will present a novel example while using intense femtosecond laser pulse to generate this kind of solitons in air.

Considering interactions between two separated in-phase and out-of-phase MABs that are sketched in Fig. 5, we may achieve breathing-like solitons in air. The main distinction between in-phase and out-of-phase cases is whether the symmetrical counterpart is imposed a$\pi $phase shift, while they have the same initial intensity distribution. We perform the simulation by using the initial profile in the x direction as illustrated before and change the initial peak intensity to${I}_{0}=10\text{\hspace{0.05em}}\text{TW}/{\text{cm}}^{\text{2}}$. Strictly speaking, a soliton must be long-lived during propagation. In the in-phase and out-of-phase cases, the structure of the high intensity distributions may not propagate a distance long enough to be define as soliton due to the relative low input power. For simplicity, we just name it as quasi-soliton and the lengths of them can be prolonged with higher input power.

Figure 6(a) shows the 2D representation of the intensity evolution for the in-phase case at transverse plane$(x=0,y)$as a function of propagation distance$z$. Apparently, benefit from the accelerating property of MABs all the lobes will self-focus into the center gradually. Two main lobes meet with each other at$z=0.55\text{\hspace{0.05em}}\text{m}$forming an intense sharp peak at the onset of the focus. This filament generated by main lobes would propagate straightly in the central area until the secondary lobes blend into the center at$z=0.75\text{\hspace{0.05em}}\text{m}$and then the second intense sharp peak is formed. Then, with sufficient energy replenished by satellite lobes merging at the central area, a quasi-soliton is constructed propagating straightly. When the degree of nonlinearities is increased to a proper level, the self-focusing, defocusing and diffraction effects are balanced, which leads to the formation of uniform quasi-solitons in a certain propagation distance. However, the quasi-soliton will become weaker with further propagation as the intensity decreases which can weaken the attractive force caused by Kerr nonlinearity. In previous works, the topic of fusing multiple filaments has received extensive interest [48–50]. The fusing process occurs when the initial beam is focused by a modulation or within the Kerr-interaction distance. However, the merging phenomenon taking place for the in-phase MAB is due to the accelerating nature. Besides, after merging the main lobes, then the secondary lobes would merge, one by one, and some part of energy acts as an energy reservoir. Figure 6(c) shows the 2D representation of the intensity evolution for the out-of-phase case. Similar to the former case, each lobe and its counterpart also would self-focus into the center. The most notable distinction is that they never merge as one filament. On contrary, when the lobes are accelerated to the center to meet their counterparts, they are bounced back toward outside. However, the Kerr nonlinearity can really attract them together. Consequently, the breathing fashion of propagation in air is formed between the distances at$z=0.5\text{\hspace{0.05em}}\text{m}$and at$z=1.1\text{\hspace{0.05em}}\text{m}$. To be more specific, the breathing quasi-soliton pairs propagate in the separated trajectories. The narrowest parts of the trajectories look like the bottlenecks, where the two separated parts get close to each other, but they will never overlap. In this part, the out-of-phase MAB does not contain a vortex since the profile in the x direction is adopted as a Gaussian function.

The difference between these two unusual behaviors can be well explained by the phase distributions and optical fields’ interference. The phase distributions in Figs. 6(b) and 6(d) show a great distinction in the two sides of red dashed line. As shown in Fig. 6(b), the fields in two side areas near the line possess the same phase. Nevertheless, for the out-of-phase case, the fields in two side areas near the line always exhibit a$\pi $phase shift [Fig. 6(d)]. The phase difference leads to destructive interference when the counterpart lobes both converge towards the propagation axis. As a result, an intensity minimum on the propagation axis is observed. The quasi-soliton split by the phase wall containing two parts can also be called dipole-mode vector quasi-soliton. Since satellite lobes have a strong contribution to the replenishment of energy toward the center leading to a high level of intensity, during propagation two parts would couple with each other and be trapped jointly by the nonlinearities of filaments.

These intriguing phenomena were also been reported in former works with the help of special nonlinear media, but not in air. During recent years, the propagation properties of the self-accelerating beams in nonlinear media are the subjects of intense research. It is found that spatial solitons can be formed with the Airy beams in nonlinear media [51, 52], which come from the balance between the nonlinear self-focusing and linear diffraction. Similar to this work, many works have shown that the bound states of spatial solitons can be generated from the interactions between Airy beams [53–56] and stationary bound states of breathing Airy soliton pairs can also been formed in nonlocal nonlinear media with proper control of the initial parameters [57]. In general, nonlocal nonlinearity means the light-induced refractive index change of a material at a particular location is determined by the light intensity in a certain neighborhood of this location [53] in optical domain. However, it has been reported that the nonlocality that can have great influences on the interaction dynamics of self-accelerating beams existing mainly in nematic liquid crystals [58] and thermal media [59]. Therefore, forming these two kinds of solitons in air without the assistance of hard-making media may extend their potential applications.

Since the plasma plays the key role in forming a self-guided structure for filament, it is important to monitor plasma channels excited by ionization of air molecules, as shown in Fig. 7 for the electron density level of${10}^{19}{\text{m}}^{-3}$. According to the slice-by-slice self-focusing mechanism of filamentation, the front part becomes narrower and narrower with further propagation because it undergoes multiphoton absorption and thus generates plasma behind. The back part would interact with the plasma leaved behind by the front part, and then it becomes “flat” which is referred to background energy reservoir. Before the end of the filament, this part of energy propagates together with the core without diffracting out. Therefore, the plasma ionized form oxygen molecules can greatly influence the propagation patterns. In Fig. 7(a), it is visible that a widely plasma-distributed area exists while the first and second intense sharp intensity peaks are formed between the distances at$z=0.6\text{\hspace{0.05em}}\text{m}$and$z=0.9\text{\hspace{0.05em}}\text{m}$. The high level of plasma density caused by the highest intensity [Fig. 4(a)] in this area can adequately defocus part of energy out the core, resulting in the loss of energy of the filament propagating straightly. The discontinuous plasma channel acts as a counterpart to overcome the Kerr nonlinearity and therefore a self-guided structure is formed. For the out-of-phase case in Fig. 7(b), the breathing fashion of plasma channels is also constructed. The area at roughly$z=0.9\text{\hspace{0.05em}}\text{m}$contains the highest level of plasma. So, the energy toward outside at this distance is mainly caused by both the phase wall and plasma defocusing effect. In other words, since the plasma is mostly accumulated near the propagation axis, which is harmful for the Kerr nonlinearity to attract the energy toward the beam center, the breathing fashion of dipole-mode vector soliton may be destroyed by plasma to some extent.

To get a deeper insight into the role of filamentation in forming spatial quasi-solitons, we compare the transverse energy flux distributions for these two special cases. At the beginning of propagation, two axial symmetrical tails of lobes accelerate toward the propagation axis, showing no significant difference [Figs. 8(a) and 8(e)]. During this stage, the linear diffraction would govern the accelerating actions of the beams since the intensity is not high enough to induce strong nonlinearities even though each single lobe may shrink gradually like the Airy beams. Upon further propagation, when the spacing between two main lobes close to each other within the Kerr interaction distance at about$z=0.5\text{\hspace{0.05em}}\text{m}$, the in-phase lobes converge together as an trajectory faster than the predicted elliptical trajectory but the out-of-phase ones shows a slower acceleration speed, shown in Figs. 6 and 7. It is evident that which kind of interference, instructive or destructive, on the propagation axis would influence the energy evolution of the quasi-solitons. When instructive interference occurs in the beam center, the on-axis intensity would increase sharply causing the Kerr nonlinearity strongly attract the lobes energy. Otherwise, the beam center has a minimal intensity and thus the Kerr nonlinearity attracts the periphery energy toward each lobe’ center respectively, which gives rise to a repellent force. Consequently, in-phase main lobes blend into one as a single filament [Figs. 8(b) and 8(c)] whereas out-of-phase main lobes are constantly split by a tiny gap [Figs. 8(f) and 8(g)]. Finally, the in-phase quasi-soliton will propagate as the form of single filament while for the out-of-phase quasi-soliton the two parts will couple with each other through Kerr nonlinearity and propagate almost separately. The results of the out-of-phase case is more intriguing, because under the impact of strong Kerr effect when two main lobes extremely close to each other, as shown in Fig. 8(h), it seems no energy exchanges between them. It gives the evidence that the phase wall with$\pi $phase shift is capable of cutting off energy flux and hence the coupling between two parts is merely due to special gradient of refractive index caused by Kerr nonlinearity. The principle of why the energy flux is cut by the phase jump is similar to why femtosecond laser filament array can be generated by the step phase plate in air [39].

Light bullet formation is another area that has been studied widely and extensively, and it is one of phenomenon accompanying filamentation for the Gaussian pulse case [60, 61]. The intention of this part is that we transform the autofocusing beam into accelerating ring-MABs able to act as light-bullet wavepackets propagating extended distances. As represented in Figs. 9(a) and (b), a handful of discrete sharp intensity peaks in the linear regime may be transformed into numerous periodic intensity peaks existing for a long distance in the nonlinear regime. Here, we construct the ring structure by rotating case 2 in Fig. 1(b) and adopt the initial peak intensity as$0.8\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{TW}/{\text{cm}}^{\text{2}}$. In the linear regime, we perform our simulation by removing the Kerr nonlinearity and multiphoton ionization (MPI), whose result is demonstrated in Fig. 9(c). In this case, the only one peak up to$90\text{\hspace{0.05em}}\text{TW}/{\text{cm}}^{\text{2}}$takes place at$z=0.75\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{m}$when the two main lobes converge with each other. It can be understood easily that the main lobe converges together at$z=0.75\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{m}$and instructively interfere with. In disappearance of the nonlinearities, the intensity can be overlapped up to$90\text{\hspace{0.05em}}\text{TW}/{\text{cm}}^{\text{2}}$by converging the energy of the main lobe together at the position of sharp peak. When this beam comes to the nonlinear regime, it is very impressive that a chain of intensity peaks is generated successfully and elegantly along the propagation axis. It is worth noting that during the gap between adjacent intensity peaks, a well-established energy reservoir with relative low intensity surrounds the core. This is of key importance to sustain the filament because the Kerr nonlinearity would attract it back to the core and then another clamping intensity peak is formed. Moreover, we can see clearly that the background energy mainly supported by the satellite lobes constantly moves toward the core during propagation. One may also find that the period of intensity peaks become longer when the intensity is lower. This is because the nonlinear dynamics start to weaken and linear effects, such as diffraction and dispersion, tend to dominate the evolution process. The collapse of ring-Mathieu beam can be divided into three stages: (i) a quasi-linear stage before$z=0.5\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{m}$during which the lobes follow an elliptic trajectory toward the propagation axis; (ii) a second stage is from$z=0.5\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{m}$to$z=1.0\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{m}$, during which the intensity increases abruptly but in a way significantly different from the linear case. In the nonlinear regime, the first intensity peak takes place at$z=0.6\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{m}$with$0.1\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{m}$more previous than that of the linear case due to the Kerr nonlinearity attracting part of the energy of the main lobe. Upon further propagation at roughly$z=0.7\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{m}$, the secondary lobes converge together with the previous energy resulting in a wide intensity plateau; (iii) a third stage is that the energy of the wave packet is merged into a single filament propagating along the propagation axis that is encompassed by the energy reservoir.

A complex phenomenon worth mentioning is that the adjacent lobes exist a$\pi $phase jump. As we can see clearly in Figs. 9(c) and 9(d), the satellite lobes would change their predesigned trajectory, decrease their acceleration and turn into a fashion of propagating straightly when closing to the beam center. The reason behind it is as same as the case of breathing quasi-soliton mentioned above. This unique property is greatly different with the filamentation of Gaussian beam whose energy all collapse toward the core and then is only defocused by the plasma. However, for the ring-Mathieu beam, the phase jump is able to prevent the satellite lobes from collapsing into the core, leading them merely acting a role of energy reservoir. Readily, this kind of energy reservoir can remain surrounding the filament core with an extended propagation distance due to the diffraction-free nature of ring-Mathieu beams.

To understand how the transverse energy influx of a ring MAB influences the reshaping process better, the spatiotemporal intensity distributions are monitored in Fig. 10. The spatiotemporal dynamics of ring MAB show great difference when compared with that of Gaussian beams which have been extensively studied in previous works [31, 62]. For typical spatiotemporal reshaping process, the pulse front generates plasma that leads to severe defocusing of the back part. Then, multiphoton absorption of the front edge gradually decreases the intensity and the back part will self-focus due to Kerr nonlinearity. As a result, pulse splitting occurs and ultimately the pulse front will be exhausted. However, the dynamics of ring MAB hold a different mechanism because no pulse splitting is observed in temporal domain during dozens of centimeters after the first focus. As shown in the top row in Fig. 10, right after the filament is formed, there is no doubt that the front is weakened by mutipoton absorption and a portion of energy in the back is defocused out. In this case, the core is continuously fueled by the energy flux from the satellite lobes of ring MAB. In the next stage of filamentation, the dynamics are featured by the mechanism similar to the typical reshaping process of Gaussian beam, as shown in the bottom row in Fig. 10. This lies in the fact that satellite lobes’ energy is no longer attainable sufficiently because their energy mostly focuses into the position around $z=0.75\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{m}$[Fig. 9(c)]. In the post filamentation stage, the wave packet or the quasi-soliton spread owing to linear diffraction.

## 4. Conclusion

In summary, we propose a scheme of laser filamentation in air via Mathieu modulation that forms curved filaments, in-phase and out-of-phase quasi-solitons and nonlinear light bullets. To be more specific, accompanied by nonlinearities, an intense MAB could generate a filament propagating along a predesigned elliptical trajectory. The size of the main lobe becomes smaller for higher ellipticity, leading to less background energy of the filament generated by main lobe, which is harmful for the elongation of curved filaments. Then, we consider interactions between two separated in-phase and out-of-phase MABs that form different kinds of quasi-solitons in air. For the in-phase case, the main lobes and secondary lobes meet their counterpart, respectively, in the center, resulting in two sharp intensity peak. Upon further propagation, sufficient energy replenished by satellite lobes instrumentally benefits the construction of a quasi-soliton propagating straightly. For the out-of-phase case, the phase wall split the filaments acting as a dipole-mode vector quasi-soliton. During propagation, they would couple with each other and be trapped jointly by the nonlinearities of filamentation in a breathing fashion. Last, we attempt to form a light bullet in air using intense ring MAB. It is very impressive that a chain of intensity peaks is generated successfully and elegantly. This is mainly because the satellite lobes that constantly move toward the core during propagation can support the background energy. According the analysis of spatiotemporal dynamics, we can divided the evolution of filament into two stages. During dozens of centimeters after the first focus, the core is continuously fueled by the energy flux from the satellite lobes of ring MAB, causing the disappearance of pule splitting. In the next stage, the dynamics can be explained by the mechanism of the typical reshaping process for Gaussian beam.

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