1 Introduction
Airy beams in free space or in linear dielectric media have become an active research topic and many literature associated with this subject were published in the past decade,[1-9] the intriguing features of which were revealed as self-acceleration, self-healing, and nondiffraction over many Rayleigh lengths. Among a quiverful of the previous literatures on Airy beam, various three-dimensional (3D) spatiotemporal wave packets or light bullet were constructed by Airy pulse in combination with other nondiffracting field configurations. They may arise in dielectric media for a robust balance between diffraction/dispersion and medium's nonlinearity. The generation of light bullet is a nontrivial task from the analytical and numerical points of view, and even more complicated in real experimental settings. For example, Abdollahpour et al. have introduced the spatiotemporal Airy light bullets composed by Airy pulse in time with two spatial Airy beams.[10] To date, research on linear light bullets as solution of the 3D potential —— free Schrödinger equation, including spatiotemporal Airy-Bessel light bullets,[11] Airy-Laguerre-Gaussian (AiLG)[12] wave packets, Airy-Hermite-Gaussian (AiHG) beams[13] localized wave peckets, Airy-Kummer-Gaussian (AiKG)[14] and Airy-Ince-Gaussian (AiIG)[15] light bullets, have been reported. Owing to without external potential, the reported spatiotemporal wave packets would retain their energy features over several Rayleigh lengths and other properties in free space. In addition, study of Airy light bullet is still an active topic, and recently the evolution dynamics of nonlinear Airy light bullets, and vortex-like and the Guassian-like Airy wave packets in self-defocusing kerr medium have been reported in Refs.[16-17].
Here we are specifically concerned with what happens in an external parabolic potential —— a workhorse of the introductory quantum mechanics.[3] The external parabolic potential is used in but not limited to Bose-Einstein condensates, laser-plasma physics, ultracold atoms, ion-laser interactions, and optical lattices. In this work, we simulate the propagation properties of Airy-Bessel light bullets in 3D Schrödinger equation with a self-consistent trapping potential, which has attracted little attention to the best of our knowledge. The light bullet used in the article is constructed by Airy pulse in time and a spatial 2D Bessel beam (not nonlinear mode solution), which will be the input when we execute the propagation. We demonstrate that not only the form and topological charges (TCs) of the incident light bullet but also the external parabolic potential greatly affect the evolution of Airy-Bessel light bullets in the medium. Its profile undergoes a profound change during propagation, which is quite different from the case in free space.
2 Theoretical Nonlinear Model and the Airy-Bessel Light Bullets
$\text{i}\frac{{\partial \psi }}{{\partial \zeta }}+ \frac{1}{2}\Bigl( {\frac{{{\partial ^2}\psi }}{{\partial {R^2}}}+ \frac{1}{R}\frac{{\partial \psi }}{{\partial R}}\!+\! \frac{1}{{{R^2}}}\frac{{{\partial ^2}\psi }}{{\partial {\varphi ^2}}}\!+\! \frac{{{\partial ^2}\psi }}{{\partial {T^2}}}} \Bigr) \!-\!\frac{1}{2}\alpha {R^2}\psi=0, $
in Eq. (1), the complex envelop of the beam and width of the potential, usually coming from an appropriate change in the medium's refractive index, are represented and controlled by $\psi \left( {\zeta, R, \varphi, T}\right)$ and $\alpha$, respectively. We normalize coordinates $T=t/t_0$ and $R=\sqrt {{X^2} + {Y^2}}$ with $X=x/w_0$ and $Y=y/w_0$. Here, the variables ${w_0}$ and ${t_0}$ are corresponding to the initial beam width of Gaussian beam and temporal scaling parameter, respectively. The propagation distance is normalized by $\zeta=z/\zeta _R$ with Rayleigh length ${\zeta _R}=kw_0^2$ and the wave number $k=2\pi/\lambda_0$ (${\lambda _0}$ is the vacuum wavelength).
Without considering the last term $\alpha R^2/2$, the solution of Eq. (1) can be written in the form $\psi \left({\zeta, X, Y, T}\right)=\phi(\zeta, T)N(R)\Phi (\varphi)$. After plugging this ansatz into Eq. (1), we end up with the following equations
$\text{i}\frac{{\partial \phi }}{{\partial \zeta }} + \frac{1}{2}\frac{{{\partial ^2}\phi }}{{\partial {T^2}}} + \beta \phi = 0,$
$ \frac{{{\partial ^2}\Phi }}{{\partial {\varphi ^2}}} + {m^2}\Phi {\rm{ = }}0,$
$\frac{1}{2}\Bigl( {\frac{{{\partial ^2}N}}{{\partial {R^2}}} + \frac{1}{R}\frac{{\partial N}}{{\partial R}} - \frac{{{m^2}}}{{{R^2}}}N} \Bigr) - \beta N = 0,$
$\psi ( \zeta = 0,R,\varphi ,T )=[ {{\phi _1}( {\zeta = 0,T} ) + {\text{i}}{\phi _2}( {\zeta = 0,T} )} ] \\ \times N(R) \Phi( \varphi ),$
where
${\phi _ \pm }( {Z{\rm{ = }}0,T} ) = \text{Ai} ( {\sigma T} ){{\text{e}}^{\sigma aT}},$
$ N( R )={J_m}( R ),$
$ \Phi ( \varphi )=[\cos(m\varphi ) + {\text{i}}q\sin (m\varphi)],$
where the Ai$(\cdot)$ is Airy function and $a$ is the decay parameter —— a positive real number ($0 < a < 1$) that makes the total energy finite[22-23] and beam more easily fit into the potential well. The parameters $\sigma =1$ and $\sigma = - 1$ for Airy beam are related to the self-accelerating and decelerating with propagation distance in the positive $T$ direction, respectively. ${J_m}\left( R \right)$ represents the Bessel function of the first kind of the order $m$. From Eq. (3), it is evident that the spatiotemporal structure of the light bullet is characterized by the TCs $m$, the modulation depth $q$ ($0 \le q \le 1$), and the decay parameter $a$. We will present that the typical dynamics of such optical field can be easily controlled through adjusting the three parameters.
We present our results in Figs. 1-5. Taking $a{\rm{ = }}0.1$ for instance, the intensity profiles of the finite energy Airy pulses for ${\phi _ + }\left( { T} \right)$ and ${\phi _ - }\left( { T} \right)$ are shown in Fig. 1. One can see that with propagation distance the acceleration of the pulse performs in the positive (negative) $T$ direction for $\sigma =1$ ($\sigma = - 1$). Clearly, attenuating slowly upon propagation, the individual pulse characterizes a long flat tails, which enables a physical realization of the beam with finite energy. For the reason that the propagation properties of the self-accelerating Airy beam have been extensively investigated both in theory and experiment,[1, 2, 5, 24] we only focus our attention on the self-decelerating Airy beam with $\sigma = - 1$ in the following.
Fig. 1 (Color online) The intensity distributions of the finite energy Airy pulses for ${\phi _ + }\left( {T} \right)$ and ${\phi _ - }\left( {T} \right)$ for the decay parameter $a = 0.1$. |
Fig. 2 (Color online) Contour plots of the optical field $N\left( R \right)\Phi \left( \varphi \right)$ distributions for some specific values of the TC $m$ and modulation depth $q$: (a) $m = 0$; (b) $m = 1$, $q = 0.5$; (c) $m = 1.5$, $q = 0.5$; (d) $m = 1.5$, $q = 1$. |
Fig. 3 (Color online) Isosurface intensity contour plot of the fundamental self-decelerating Airy-Bessel light bullet with $m = 0$ at $\zeta = 0$ (top row) and $\zeta = 5{\zeta _R}$ (bottom row). Figures in the column right present the vertical view in the $T = 0$ plane from directly above. The blue arrow depicts the direction of acceleration. |
Fig. 4 (Color online) Vortex Airy light bullets with, $m = 1$ and $q= 1$. Setup is the same as in Fig. 3. |
Fig. 5 (Color online) Snapshots describing the evolution of symmetric and asymmetric necklace Airy-Bessel light bullets at $\zeta = 0$ (the left column) and $\zeta = 5{\zeta _R}$ (the right column) with $m = 1$ (top row) and $m = 1.5$ (bottom row). |
The common characteristic intensities of $N\left( R \right)\Phi \left( \varphi \right)$ modes with varying $m$ and $q$ are presented in Fig. 2. For the case shown in Fig. 2(a), the amplitude distribution features obviously azimuthal symmetric owing to azimuthal angle $m = 0$. If $m$ is increased to 1, the corresponding number of azimuthal node will form to make the concentric rings transform into azimuthally-modulated shapes, characterizing $2m$ beads, as displayed in Fig. 2(b). With $m=1.5$, Fig. 2(c) also complies the rule, while intensity distribution becomes asymmetric compared to the two former cases. Now we set $q = 1$ and $m = 1$, and redo the simulation presented in Fig. 2(d); in comparison with the result shown in Fig. 2(a), we find that the beam profile features ring-like structure and appears one singularity around the origin.
3 Analysis and Discussion
To study the formation and dynamics of self-decelerating Airy-Bessel light bullets carrying different TCs $m$, we solve propagation Eq. (1) numerically by the fourth-order split-step fast Fourier transform (FFT) method in double precision. As initial condition, we take the expression (3) of multicharged beam perturbed by 10% of noise. To make beams of finite energy and prevent FFT spillover effects, we use an aperture with a diameter large enough to enforce fast convergence of beam intensity to zero at the transverse infinity. In the following numerical example, we set the relevant parameters to $a = 0.1$, ${\omega _0} = 100 \mu $m, $\lambda=530$ nm, $\alpha=1$ and propagation distance $\zeta = 5{\zeta _R}$.[12, 13, 21]
3.1 Zero-Vorticity Airy-Bessel Light Bullets $(m=0)$
Figure 3 displays an example of self-decelerating Airy-Bessel light bullet with $m = 0$ in an external potential. The isosurface plots illustrate the full 3D evolution of the light bullets at $\zeta = 0$ (the top row) and $\zeta = 5{\zeta _R}$ (the bottom row) while the right column depicts intensity distribution in the $T = 0$ plane. As presented by the 3D plot of incidence light bullet, combination of the fundamental Gaussian state with the finite-energy Airy pulse, it is characterized by ellipsoidal pulses stacked in the vertical direction and appearance of two coaxial ring around the first pulse. Owing to $m = 0$, there do not exist necklace ellipsoids. Obviously, from the isosurface intensity at $\zeta = 5{\zeta _R}$, we find that the pulse stacked of light bullet become compressed and their shape suffer significantly change during propagation. This phenomenon originates from the phase transition that an Airy profile is transformed into a Gaussian one at a critical point upon propagation due to the manipulation of induced potential.[16, 25-27] It is a consequence of the beam approaching and bouncing off the potential wall. The beam looses its multi-peak profile and propagates as a symmetric Guassian profile. This behaviour is clearly different from the results obtained in free space,[12-15, 21] in which the structure of such wave packet propagates over Rayleigh lengths without appreciable change in their properties. In addition, it should be emphasized that the pulse interval is crucial difference between $\zeta = 5{\zeta _R}$ and $\zeta = 0$. Specifically, the pulse interval ranges from $-10$ to 10 in former case, while in later case it ranges from 0 to 8. As expected, this indeed shows that the light bullet is decelerating, which is accelerating in the direction opposite to the blue arrow. These results are similar to the results presented in Ref. [26].
3.2 Vortex Airy Light Bullets ($q = 1,m \ne 0$)
Let us focus on the case in which the vortex-shaped field structure is constructed with $q = 1$ and $m{\rm{ = }}1$. As shown in Fig. 4, the top isosurface plot represents the initial beam profile at $\zeta = 0$. In comparison with those shown in Fig. 3, we find that the central ellipsoidal pulses change into vortex rings. The underlying physics in this setup is fundamentally different from the previous setup. The term $\cos \left( {m\varphi } \right) + {\text{i}}q\sin \left( {m\varphi } \right) = {{\text{e}}^{{\text{i}}\varphi }}$ is fulfilled in Eq. (3) under the condition $q = 1$ and $m =1$. Thus, the phase distribution appears one singularity (called the vortex core) and energy flow around it, while the intensity is equal to zero at singularity point.[26] This is for the reason that along $T$ axis the intensity structure is characteristic of vortex rings, as depicted in second subplot of Fig. 4. The patterns at propagation distance $\zeta = 5{\zeta _R}$ are displayed in the bottom row. We find that it shares the dynamical behaviour of former case with $m = 0$ and can also be explained by the reason discussed before.
3.3 Necklace Airy-Bessel Light Bullets ($q = 0,m \ne0$)
Last, but not the least, the necklace Airy-Bessel light bullets can be obtained by setting parameters to be $q = 0$ and $m \ne 0$ in Eq. (3), as illustrated in Fig. 5. Owing to the number of TC $m \ne 0$, the disk-shaped fundamental pulse in each layer are transformed into necklace form with $2m$ beads, and the intensity structure involves 3D necklace beams.
More specifically, as presented in the first subplot of Fig. 5, for $m = 1$, the structure is fragmented with five layers in T-axis, and two ($2m$) necklace ellipsoids in the ($x, y$) plane in each layer. It comes from the TC dependence in Eq. (3). The second subplot displays the pattern of numerical simulation at $\zeta = 5{\zeta _R}$. The manipulation of the induced potential also results in profound shape change that in the vertical direction the necklace light bullet feature distinct compressed and more layers are appeared with increase of propagation distance.
In Fig. 5, if TC $m$ is set as half-integer and limit $q = 0$, we do the corresponding simulations and the results are shown in the bottom row, i.e., an example of the asymmetric necklace pattern. Compared with the results obtained in the former case, one observes asymmetric 3D necklace-shaped wave packets that asymmetric single-layer and multilayer necklace structures in each layer are featured in T-axis, as displayed in bottom row with $m = 1.5$. In addition, we see that the phenomena observed and laws formulated before are still applicable.
4 Conclusion
In a linear medium with a self-consistent trapping potential, we have investigated numerically the dynamics of self-decelerating Airy-Bessel light bullets carrying different TC, exhibiting fundamental zero-TC pulsed structures or non-zero-TC vortices. We have demonstrated that the spatiotemporal profiles can be significantly changed in induced potential during propagation, which is quite different from that in free space. The appearance of disk-shaped fundamental pulses, the necklace rings and the vortex rings results from a combined action of the number of TC, modulation depth and induced potential in the linear medium, all of them acting simultaneously. Our approach may not only prove useful for understanding self-accelerating beams, but also lead to potential applications in signal processing, particle manipulation, and other fields.