## Abstract

Three-dimensional chirped Airy Complex-variable-function Gaussian vortex (CACGV) wave packets in a strongly nonlocal nonlinear medium (SNNM) are studied. By varying the distribution parameter, CACGV wave packets can rotate stably in a SNNM in different forms, including dipoles, elliptic vortices, and doughnuts. Numerical simulation results for the CACGV wave packets agree well with theoretical analysis results under zero perturbation. The Poynting vector related to the physics of the rotation phenomenon and the angular momentum as a torque corresponding to the force are also presented. Finally, the radiation forces of CACGV wave packets acting on a nanoparticle in a SNNM are discussed.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Over recent decades, the generation and manipulation processes of stable three-dimensional (i.e., localized in both space and time) wave packets have presented a number of fascinating challenges. One major problem in optics is that, under the combined actions of diffraction and dispersion, these wave packets tend to broaden in both space and time [1,2]. Wave packets such as the Bessel and Airy types, which will propagate in a nondiffracting manner without any requirement for equalization of the diffraction and the dispersion, have thus attracted particular attention [3–6]. By combining Bessel beams in the transverse plane with temporal Airy pulses, Chong et al. reported observation of a versatile class of spatiotemporal linear light bullets [7]. Abdollahpour et al. realized intense $Airy^3$ (involving Airy distributions in both the spatial and temporal dimensions) light bullets in both the linear and nonlinear regimes [8]. The generation of optical $Airy^3$ bullets with spatiotemporal optimal compression and energy confinement, while still maintaining their self-accelerating and self-healing properties, has been a subject of further research [9]. Additionally, the characteristics of many types of localized Airy-related wave packets in linear media, including self-accelerating Airy-Laguerre-Gaussian [10], Airy-Ince-Gaussian [11], Airy-Tricomi-Gaussian [12], Airy Gaussian vortex [13], and dual-Airy-ring Airy Gaussian vortex [14] wave packets have been discussed because of their special characteristics. In fact, the Airy function provides a unique dispersion-free solution in one-dimensional spatiotemporal wave packets generated using an Airy pulse in combination with another nondiffracting field, thus opening up exciting possibilities for use in numerous applications, ranging from optical tweezing to plasma physics [8,9,15–22].

Wave packets propagating in nonlinear media have been studied extensively in recent years. Nonlocality has provided new physical effects in the nonlinear regime, and experimental observations of nonlocal spatial optical solitons in both lead glasses [23,24] and nematic liquid crystals [25,26] have been reported. In the case of strong nonlocality, where the characteristic length of the material response function is much greater than the beam width, Snyder and Mitchell simplified the nonlocal nonlinear Schrödinger equation to derive the Snyder-Mitchell model [27]. Therefore, tremendous research efforts have been made to obtain wave packets in a strongly nonlocal nonlinear medium (SNNM). A novel class of Complex-variable-function Gaussian beams [28] and solitons [29] obtained by using different input powers was introduced by Deng et al. In addition, ellipticons, Airy beams, and vortex beams have been studied extensively in the SNNM [30–36]. Recently, 3D localized Airy-Laguerre-Gaussian wave packet propagation in a SNNM has been investigated [37]. However, the propagation of chirped Airy Complex-variable-function Gaussian vortex (CACGV) wave packets is very different.

Surprisingly, this problem has not been considered previously in the research literature to date. In view of the importance of Complex-variable-function Gaussian vortex (CGV) and Airy-related spatiotemporal wave packets, the topic of CACGV wave packets requires in-depth study. Here, we investigate the propagation and radiation force properties of CACGV wave packets in a SNNM, both theoretically and numerically.

The rest of this paper is organized as follows. In section 2, we introduce the $(3+1)$-dimensional nonlinear evolution equation for optical wave packets in rectangular coordinates, and obtain two types of localized CACGV wave packets, i.e., the chirped Airy cosine Complex-variable-function Gaussian vortex (CAcosCGV) and the chirped Airy sine Complex-variable-function Gaussian vortex (CAsinCGV), in a SNNM. In section 3, we research the effects of the distribution parameter $b$, the first-order chirp parameter $c_{1}$ and the second-order chirp parameter $c_{2}$ on the CACGV wave packets. Then, related numerical simulation snapshots of CACGV wave packets under various perturbation conditions are discussed. The rotating Poynting vector and the angular momentum are also presented. In section 4, the radiation force of the CACGV wave packets acting on a Rayleigh dielectric particle is studied. Finally, we present our conclusions in section 5.

## 2. The CACGV wave packet model in a SNNM

The propagation of a spatiotemporal wave packet in a nonlinear medium obeys the $(3+1)$-dimensional nonlinear Schrödinger equation [37,38]

In the dimensionless coordinate system $(X, Y, T, Z)=(x/w_{0}$, $y/w_{0}$, $t/t_{0}$, $z/L_{diff})$, $t_{0}$ is the temporal scaling parameter, $L_{diff}=kw_{0}^{2}$ is the diffraction length, and $L_{disp}=t_{0}^{2}/\left | k_{2} \right |$ represents the dispersion length. We assume that both the dispersion and the diffraction have the same effect along the propagation direction (i.e. $L_{diff}=L_{disp}$). In the strongly nonlocal case, where $w_{m}\to \infty$, the length of the wave packet is very short when compared with the length of the response function. When the response function $R$ expands to the second order, Eq. (1) can be simplified to give the normalized dimensionless Snyder-Mitchell model [27,37]

where $\nabla ^{2}=\partial ^2/\partial X^2+\partial ^2/\partial Y^2+\partial ^2/\partial T^2$ is the spatiotemporal Laplacian operator, $\eta =P_{0}/P_{c}$ is the power ratio, $P_{0}$ is the input power at $Z=0$, $P_{c}=n_{0}/(\gamma n_{1}L_{diff}^{2})$ is the critical power for the soliton propagation, and $\gamma$ is a material parameter related to $R$. Then, using the method of separation of variables, we assume a solution of the following form By substituting Eq. (3) into Eq. (2), the following two equations are obtainedThe initial CGV wave packet can be expressed as

## 3. Propagation properties of the CACGV wave packets in a SNNM

#### 3.1 Localized CACGV wave packets in a SNNM

Figure 1 shows images of the spatiotemporal localized CAcosCGV wave packets, and the transverse intensity and phase distributions for various values of $b$ at the initial plane $Z=0$. Interestingly, when $b$ is varied, the CACGV wave packets take different forms, including dipoles (Fig. 1(a1)), an elliptical vortex (Fig. 1(a2)), and a doughnut (Fig. 1(a3)). It is clearly shown that the CAcosCGV wave packets cluster more closely around the center of the wave packet as $b$ increases and approach a transverse Gaussian vortex distribution when $b\to \infty$, because $\cos [(X+iY)/b]=1$ when $(X+iY)/b=0$. In Figs. 1(a2)–1(c2), the center of the intensity distribution of the CAcosCGV wave packet is slightly lower than the intensity in its immediate surroundings because of the vortex factor. The related vortex distributions are shown in Figs. 1(a3)–1(c3).

Figure 2 illustrates the analytical solution to Eq. (10) at various propagation distances with different chirp parameters. The spatiotemporal CAcosCGV wave packets rotate as a whole as the propagation distances increase, thus maintaining the positioning of their main lobes at the front, and they undergo positive acceleration with the positive first-order chirp parameter $c_{1}$. Surprisingly, it is found that the CAcosCGV wave packets are reversed and have their side lobes at the front with the negative second-order chirp parameter $c_{2}$. From Eq. (6), the temporal ballistic trajectory of these CACGV wave packets can be expressed as $Z^{2}/(4+8c_{2}Z)+c_{1}Z$. When $c_{1}=c_{2}=0$, the trajectory returns to that of the initial concept of self-accelerating wave packets without the requirement for an external potential [3]. Here, the first-order chirp parameter $c_{1}$ will speed up (positive $c_{1}$ in Figs. 2(d)–2(f)) or slow down (negative $c_{1}$ in Figs. 2(a)–2(c)) the acceleration, while the negative second-order chirp parameter $c_{2}$ in Figs. 2(g)–2(i) will cause main lobe reversals. Additionally, a singularity position exists at $Z=c_{2}/2$ when $c_{2}<0$. The propagation of the wave packets involves acceleration with the heads at the front before they reach singularity position ($d>0$), while the main lobes are reversed and the tails are at front in the temporal dimension after the singularity position ($d<0$).

Figure 3 shows the spatiotemporal CAsinCGV wave packets obtained using the analytical solution given in Eq. (11), and adjusted using the first-order and second-order chirp parameters, the distribution parameter and the propagation distance. Over the same propagation distance $Z=2\pi /3$, the spatiotemporal CAsinCGV wave packets keep their heads at the front and undergo positive acceleration as shown in Figs. 3(b) and 3(e), but are then reversed to show their tails at the front in Figs. 3(c) and 3(f). The CAsinCGV wave packets rotate as stably as the CAcosCGV wave packets, except that the external overall scale is softer and fuller when $b=1.5$, and they have larger internal hollows. The CAsinCGV wave packets will approach a transverse Gaussian vortex distribution with a hollow center when $b\to \infty$, because $\sin [(X+iY)/b]=0$ when $(X+iY)/b=0$.

To perform the numerical calculations of the CACGV wave packets to verify the theoretical analysis results, we present related results in Fig. 4. We also simulate the spatiotemporal wave packets using various perturbation conditions to discuss their stability. The initial condition is supposed to be $\Psi +\delta \psi$, where $\delta$ is the perturbation parameter, and $\psi$ is a random complex function with a maximum amplitude that is less than that of the CACGV wave packets. By comparing Figs. 4(a) and 4(d) with the analytical results shown in Fig. 2(e) (for the CAcosCGV wave packets) and Fig. 3(b) (for the CAsinCGV wave packets), we find that the numerically simulation results agree well with the theoretical analysis results under zero perturbation ($\delta =0$) conditions. However, the difference between the numerical simulation results and the theoretical analysis results is huge when $\delta =0.05$, as shown in Figs. 4(c) and 4(f), while this difference becomes too small to distinguish when $\delta =0.003$, as shown in Figs. 4(b) and 4(e). A stable spatiotemporal wave packet propagates in a numerical experiment in which the initial wave packet is perturbed by noise; while this may not constitute a rigorous proof of the stability of the wave packet, it does provide strong support for the existence of observable nonlinear modes in the laboratory experiments.

#### 3.2 Poynting vector and angular momentum of CACGV wave packets in a SNNM

In addition, to determine more of the propagation properties of CACGV wave packets in a SNNM, the energy flow and the angular momentum must also be discussed. The propagation properties of electromagnetic fields are closely related to their local energy flow [40,41], which is generally expressed in terms of the Poynting vector. The Poynting vector has a magnitude expressed as the energy per unit area (or per unit time), and a direction that represents the energy flow direction at any point in the field. The vector potential is a linear polarization state. Given the polarized vector potential $\Psi (x,y,0,z)e^{-ikz}{\textbf {e}}_{x}$, where ${\textbf {e}}_{x}$ is the unit vector oriented along the $X$ direction, the time-averaged Poynting vector can then be written as $\langle {\textbf {S}}\rangle =\frac {c_{0}}{4\pi }\langle \bf {E}\times \bf {B}\rangle$ [42], where $c_{0}$ is the speed of light in free space, and $\bf {E}$ and $\bf {B}$ are the electric and magnetic fields, respectively.

The time-averaged angular momentum density for the electromagnetic field is the angular momentum per unit area (per unit time), which is obtained by forming the cross-product of the position vector with the time-averaged momentum density $\langle \bf {j}\rangle = \bf {r}\times \langle \bf {E}\times \bf {B}\rangle$ [41,42].

In Fig. 5, the directions and magnitudes of the arrows (shown as solid lines) correspond to the direction and the magnitude of the Poynting vector in the transverse plane and their values are computed numerically. The normalized angular momentum density of the CACGV wave packets is also simulated numerically, with results as shown in Fig. 6. We find that the energy flow and the angular momentum density both have a rotation angle of $2\pi$ during CACGV wave packet propagation from $Z=0$ to $Z=\pi$, which agrees with the results of rotation of the spatiotemporal CACGV wave packet distributions. The change in the Poynting vector is related to the physics of the rotation phenomenon of the wave packets, while the change in the angular momentum is a torque that corresponds to the force caused by the change in the linear momentum.

## 4. Radiation force of CACGV wave packets in a SNNM

In the following, we investigate the radiation force of the CACGV wave packets when acting on a nonabsorbent Rayleigh dielectric particle, where the particle radius $r_{0}$ is sufficiently small when compared with the wavelength (i.e. $r_{0}\leq \lambda /20$). When the particle is in the steady state, the scattering force and the gradient force are given by [43]

In a SNNM, the scattering force and the transverse gradient force of the CACGV wave packets acting on a Rayleigh particle are shown in Figs. 7 and 8. The scattering forces of both the CAcosCGV wave packets [Figs. 7(a1)–7(d1)] and the CAsinCGV wave packets [Figs. 8(a1)–8(d1)] show rotating elliptical vortex shapes that are proportional to their intensity distributions. The gradient forces of the CAcosCGV wave packets [Figs. 7(a2)–7(d2)] and the CAsinCGV wave packets [Figs. 8(a2)–8(d2)] show two loops of elliptical vortex shapes, which are given in terms of their intensity gradient. The scattering forces aim to drive the Rayleigh particles to move along the propagation direction in the SNNM, while the gradient forces pull the Rayleigh particles toward the maximum of the transverse light field. We also note that, by varing the distribution parameter, the radiation force will also be different because its intensity distribution can vary in shape from a rotating dipole (b=0.2) to a doughnut (b=10). The radiation forces still maintain their periodic oscillation properties, and the rotation ability of the CACGV wave packets provides a unique control mechanism for nanoparticles that may have important applications in optical manipulation and rotation of biological specimens [7,15,22].

## 5. Conclusion

In conclusion, by solving the (3+1) dimensional nonlinear evolution equation in rectangular coordinates, we obtained two types of localized CACGV wave packet (CAcosCGV and CAsinCGV) in a SNNM. The manipulation of these CACGV wave packets using the distribution parameter, the first-order and the second-order chirp parameters is demonstrated. It is interesting to note that the CACGV wave packets cluster more closely around the wave packet center as $b$ increases, while stable CACGV wave packets can rotate in a SNNM with different shapes, including dipoles, elliptical vortices and doughnuts. Here, the first-order chirp parameter $c_{1}$ can be used to adjust the acceleration by selecting a positive $c_{1}$ (to speed up) or a negative $c_{1}$ (to slow down), while the negative second-order chirp parameter $c_{2}$ can produce main lobe reversals. Then, the related numerical simulation snapshots of the CACGV wave packets under various perturbation conditions are discussed. The numerical simulation results agreed well with the theoretical analysis results under zero perturbation conditions. However, the difference between the numerically simulated results and the theoretical analysis results is huge when $\delta =0.05$, although this difference becomes too small to distinguish when $\delta =0.003$. Additionally, the rotating Poynting vector and the angular momentum are presented and discussed. The change in the Poynting vector is related to the physics of the rotation phenomenon for the wave packets, while the change in the angular momentum is a torque that corresponds to the force caused by the change in the linear momentum.

Furthermore, the rotating radiation forces of the CACGV wave packets acting on a nonabsorbent Rayleigh dielectric particle are also studied. The scattering forces aim to drive the nanoparticles to move along the propagation direction in the SNNM, while the gradient forces pull the nanoparticles toward the maximum of the transverse light field. We also note that variation of the distribution parameter will cause the radiation force distributions to change, with intensity distributions ranging in shape from rotating dipoles to doughnuts. With these fascinating properties, the theoretical and numerical results presented in this paper will be helpful in understanding the behavior of CACGV wave packets in a SNNM and indicate their potential applications in optical trapping of nonabsorbent nanoparticles.

## Funding

National Natural Science Foundation of China (11775083, 11947103, 61675001); Guangdong Province University Youth Innovative Talents Program of China (2018KQNCX136); Guangdong Province Nature Science Foundation of China (2017A030311025); Guangdong Province Education Department Science Foundation of China (2014KZDXM059); Guangdong Polytechnic Normal University Talent Introduction Project Foundation of China.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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