Spatiotemporal wave packet theory [1] and ultrashort laser pulses [2–7] have received considerable attention by the researchers in the field of laser technology for the past decades. Hellwarth et al. have studied the focused one-cycle electromagnetic pulses to introduce focused vector wave packets [8]. Due to the characteristic of self-acceleration [9], Airy beams propagates along a quadratic trajectory on the transverse plane, and the main lobe remains basically unchanged. The sudden autofocus ability of a circular Airy beams in a linear region has been theorized and experimentally demonstrated [10–12]. This characteristic can be used to improve the performance of optical tweezers. Richards-wolf vector diffraction integral theory can accurately describe the focal field distribution of the tightly focused vector light field [13], and has been widely used to analyze the tightly focused field of the vector light field [14, 15]. Compared with the traditional beam, the circular Airy Gaussian beam can focus automatically without lens. Due to the obvious advantages of Airy beams, such as self-healing, self-acceleration and non-diffraction, the spatiotemporal wave packet research related to Airy beams has attracted the attention of many researchers. Georgios et al. firstly proposed light bullets produced by Airy pulses [16]. Eichelkrant et al. studied the oblique spatiotemporal Airy packet in doubly dispersive optical media [17]. Zhong et al. investigated the three-dimensional finite energy self-accelerating Airy parabolic light projector [18] and the Airy-Laguerre-Gaussian wave packet [19]. Peng et al. researched the chirped Airy Gaussian vortex packets in a secondary index medium [20]. Chong et al. have achieved a three-dimensional Airy-Bessel wave packet experimentally by combining a temporal Airy pulse with a spatial Bessel beam [21]. Besides, other spatiotemporal wave packets, such as Airy-Hermite-Gaussian packets and Airy-vortex packets [22, 23] were also been demonstrated. The spatial distribution of the spatiotemporal wave packets above is scalar light field. Chong et al. proposed a universal linear spatiotemporal light wave packet called Airy-Bessel light bullet, which was first proposed as a vector light field [21]. Generating spatiotemporal wave packets that are neither affected by dispersion nor diffraction has always been a fascinating challenge. In this paper, we study the spatiotemporal autofocusing properties of the vector-Airy-circular Airy Gaussian vortex (VAi) and the vector-symmetry Airy-circular Airy Gaussian vortex (VSAi) wave packet in free space.

Since the radial symmetry of the VAi and VSAi wave packet, the Schrodinger equation in cylindrical coordinates is more suitable as following [19, 24]. i ∂ U ∂ Z + 1 2 ( ∂ 2 U ∂ R 2 + 1 R ∂ U ∂ R + 1 R 2 ∂ 2 U ∂ φ 2 + ∂ 2 U ∂ T 2 ) = 0 (1) Where, R=(X ² +Y ²)^1/2, and azimuth φ = arctan (Y/X). Here, X = x/ω ₀, Y = y/ω ₀ and T = t/t ₀ are normalized coordinates, and ω ₀ is the spatial scaling parameter, t ₀ is the temporal scaling parameter, Z is the normalized propagation distance. The wave packet can be separated as. U ( R , φ , T , Z ) = M ( R , φ , Z ) P ( T , Z ) (2)

So the Schrodinger equation can be described as. i ∂ P ∂ Z + 1 2 ∂ 2 P ∂ T 2 = 0 (3) i ∂ M ∂ Z + 1 2 ( ∂ 2 M ∂ R 2 + 1 R ∂ M ∂ R + 1 R 2 ∂ 2 M ∂ φ 2 ) = 0 (4)

The two equations describe the propagation characteristics of wave packets in time domain and space domain respectively. Based on this, we will further analyze the influence of dispersion effect on the evolution characteristics of VAi and VSAi wave packets.

Based on the analysis above, the influence of initial velocity on the evolution trajectory and the intensity of Airy pulse during the self-focusing process of circular Airy light can be analyzed following. Figures 1A–C present the evolution trajectory and Figures 1D–F show the intensity distribution of Airy pulse after different propagation distances with various initial velocities υ. The parameters adopted as ω ₀ = 1, t ₀ = 1, the temporal and spatial attenuation coefficient a _t = 0.3, the time distribution factor b _t = 0.5, the attenuation coefficient of spatial domain a _s = 0.05, the spatial distribution factor b _s = 0.2 and the radius of the initial circular Airy Gaussian light R ₀ = 2. It can be seen that the evolution trajectory of Airy pulse can be controlled by changing the initial velocity, and the initial velocity of Airy pulse has no effect on the intensity distribution during the propagation. The self-accelerating trajectory of Figure 1B,E is the same as the evolution trajectory of conventional Airy pulse for υ = 0. But when the initial velocity is negative, Airy pulse will initially propagate in the negative direction and then propagate in the positive direction for υ = -3.6 as in Figure 1A,D. When the initial velocity of the Airy pulse is in the opposite direction from the self-acceleration, it will slow down and then accelerate. Furthermore, the direction of the initial velocity of the Airy pulse is the same as the direction of the self-acceleration, and it will continue to accelerate when υ = 3.6 as in Figure 1C,F That means the evolution trajectory of Airy pulse can be modulated by its initial velocity, and the speed of the pulse will become faster versus the absolute value of the initial velocity increasing.

At the same time, the longitudinal self-focusing propagation characteristics of a radially polarized circular Airy Gauss vortex in free space when R ₀ = 2 is also analyzed. Figure 2A shows the longitudinal intensity distribution of incident light from Z = 0 to Z = 0.4. The dotted lines represent propagation positions at Z = 0, 0.1, 0.2, 0.3 and 0.4 successively. It can be found that the circular Airy Gauss vortex will focus at Z = 0.2, and the intensity peak will increase first rapidly and then decrease gradually versus the propagation distance, and the intensity maximum I _max = 5.515 at Z = 0.3.

Then the evolution of VSAi wave packet intensity during the focusing process can also be analyzed as in Figure 3. Where, I ₀ is the intensity extreme at the incident plane, and I _max /I ₀ represents the ratio of the intensity maximum during the propagation with the initial intensity extreme at the plane. It can be seen that the intensity of the VSAi wave packet can be enhanced with the symmetric incident Airy pulse pair in the time domain, and the intensity maximum at focus is about 80 times of that of the initial intensity. This provides a new idea to improve the pulse intensity at a specific position during the application of high intensity pulse processing.

Secondly, by decomposing the nonlinear Schrodinger equation and considering the third-order dispersion effect, we can get. ∂ P ∂ Z = i 2 ∂ 2 P ∂ T 2 + b 6 ∂ 3 P ∂ T 3 = 0 (5)

Where, b = β ₃/(|β ₂|b _t ) is the third-order dispersion coefficient. Then the influence of the third-order dispersion on the trajectory of the focused pulse can be analyzed by using the split step Fourier method. Figure 4 shows the influence of positive third-order dispersion coefficients on the evolution trajectory of the focused Airy pulse of T ₀ = 2 with the different third-order dispersion coefficients as (a) b = 0.1, (b) b = 0.2, (c) b = 0.3; (d) b = 0.4. It can be found that the positive third-order dispersion coefficient causes the forward propagation pulse to travel in the opposite direction, and positive third-order dispersion coefficient will cause a negative acceleration effect on the pulse. In addition, Table 1 shows the focusing position Z and the intensity peak I _max of the pulse for different positive third-order dispersion coefficients. We can see that the evolution trajectory of the focused pulse will deviate from the original trajectory versus the increasing of the third-order dispersion coefficient, the pulse will focus in advance during propagation, and the intensity peak of the focused pulse will also decrease. That’s because the increase of the third-order dispersion effect can result in the loss of pulse energy, and the decrease of the pulse energy at the focus will appear. Besides, T ₁ and Z ₁ in Figure 4A describe the change of the time and space position with the propagation pulse in the negative direction from the initial position to the focus for b > 0. Here the arctangent value of T/Z is defined as the deflection angle θ = arctan (T/Z) of propagation pulse, and it can be used to describe the influence of third-order dispersion effect on the deviation of the evolution trajectory. Figure 5A shows the variation of the pulse focusing position Z and deflection Angle θ versus the third-order dispersion coefficient b. It can be seen that the deflection angle of pulse evolution trajectory increases rapidly at first versus the increasing of b, but the growth rate tends to be flat after b = 0.4. The deflection angle of the pulse will increase by 0.2° for every 0.1 increasing of b for the case b < 0.4, but the deflection angle θ can’t reach 90° obviously, that’s because excessive third-order dispersion effecting will lead to excessive loss during the pulse propagation and failure to focus. Furthermore, the variation of the evolution trajectory deflection angle with the third-order dispersion coefficient can be approximately described as y = x ^1/2 by fitting, and the focusing position moving in the direction of Z = 0 with the increasing of the third-order dispersion coefficient. Figure 5B shows the variation of the pulse intensity maximum at the focus versus the third-order dispersion coefficient. The main peak intensity of the pulse at the focal point is closely related to the value of the wave packet equivalent surface. The intensity of the main peak at the focus of the pulse gradually decreases with increasing of b, and the intensity of the main peak at the focus is I _max = 0.2408 with b = 1. In conclusion, the deflection angle increases with the increasing of the third-order dispersion coefficients, but the focusing position will move toward Z = 0 and the intensity maximum will decrease.

Based on the analysis above, Figure 6 presents the influence of negative third-order dispersion coefficients on the evolution trajectory of the focused Airy pulse of T ₀ = 2 with the different third-order dispersion coefficients as (a) b = -0.1, (b) b = -0.2, (c) b = -0.3 and (d) b = -0.4. It can be found that the negative third-order dispersion coefficient causes the reverse propagation pulse to travel in the positive direction, and negative third-order dispersion coefficient will have a positive acceleration effect on the pulse. In addition, Table 2 shows the focusing position Z and the intensity peak I _max of the pulse for different negative third-order dispersion coefficients. We can see that the focusing position and intensity maximum are not symmetrical with positive and negative third-order dispersion coefficients.

Based on the nonlinear Schrodinger equation, we have established the spatiotemporal decoupling pulse propagation model of the vector field, and analyzed the influence of the initial time interval of focusing pulse on the pulse intensity maximum at the focus. Furthermore, we have also discussed the effects of the third-order dispersion effect at the focus and evolution trajectory of Airy pulses. It is found that the focused airy pulse with the third-order dispersion effect occupies more energy at the sub summit of the focus for the same initial intensity peak. The positive third-order dispersion coefficient causes the forward propagation pulse to travel in the opposite direction, and positive third-order dispersion coefficient will have a negative acceleration effect on the pulse. While the negative third-order dispersion coefficient causes the reverse propagation pulse to travel in the positive direction, and negative third-order dispersion coefficient will have a positive acceleration effect on the pulse. At the same time, it is also found that the intensity maximum of Airy pulse at the focus can be modulated by adjusting the third-order dispersion coefficient, and the evolution trajectory of the pulse can be accurately controlled and processed by adjusting the third-order dispersion effect. In paraxial optical system, the propagation along the Z direction in free space of spatiotemporal decoupled wave packets can be described by the normalized Schrodinger equation. Decoupling the three-dimensional (3D) wave packet as a fundamental spatiotemporal element requires a one-dimensional dispersion-free pulse, that means the wave packet can be decoupled spatiotemporal without dispersion modulation of the pulse, and the spatiotemporal computation can be carried out independently. These results can provide a theoretical basis for accurately controlling the evolution trajectory and intensity peak of the spatiotemporal self-focusing wave packet.