A type of chirped Airy Gaussian vortex (CAiGV) localized wave packets in a quadratic index medium are studied by solving the paraxial differential equation. For the first time, the propagation properties of spatiotemporal CAiGV light bullets in the quadratic index medium are demonstrated. Some typical examples of the obtained solutions are based on the temporal and spatial chirp parameters, the initial velocity, the distribution factor, and the topological charge. The radiation force of the spatial CAiGV wave packet on a Rayleigh dielectric particle has the periodically reversion and recovery abilities due to the quadratic potential. What we report here can obtain different radiation force trajectory and may have potential application in optical tweezing and bio-medical field.
© 2017 Optical Society of America
Airy wave packet has attracted extensive attention from research communities all over the world since it was introduced theoretically  by Berry and Balazs and demonstrated experimentally with finite energy [2–4] by Siviloglou and Christodoulides. The notable features of the Airy beam are self accelerating, self healing and no diffraction in the linear regime [1–5]. It was Bandres and Gutiérrez-Vega  who reported the generalized Airy Gaussian beam and analyzed its propagation through optical systems described by ABCD matrices with complex elements, and introduced it could be realized experimentally to a good approximation. According to the fact that the exponentially truncated Airy function is the Fourier transform of a Gaussian distribution with a cubic phase modulation that one can get the Airy beams [7, 8]. Therefore, the spatial Airy beam can be obtained by imposing a cubic phase on a broad Gaussian beam and then using a converging lens to perform a spatial Fourier transformation. Additionally, Deng et al [9–11] have studied Airy complex variable function Gaussian beams and the propagation of the Airy Gaussian beam in strongly nonlocal nonlinear media and quadratic-index medium. Study on the propagation of an optical vortex superimposed on Airy beams has been a highlight . Further research activities on Airy Gaussian vortex beams also have attracted high interest [13, 14], because these beams not only maintain the no diffracting propagation properties within a finite propagation distance from the Airy distribution, but also possess the phase singularity and intensity singularity from the vortex distribution.
Over the past decades, tremendous efforts have been made by researchers to generate the three-dimensional (3D) optical light bullets [15–22], which are localized in both space and time. In the nonlinear propagation regime, the generation of stable spatiotemporal light bullets is a very delicate task, since it is possible only when the effects of dispersion and diffraction are equalized. Whereas, in the linear propagation regime, Airy wave packets have been demonstrated and have opened up new exciting possibilities in numerous applications such as versatile Airy-Bessel light bullets , experimental demonstration of Airy3 light bullets , and spatiotemporal energy confinement of Airy3 light bullets . Other kinds of Airy related light bullets in free space also attracted great interest, for instance, Airy-Parabolic-Cylinder , Airy-Laguerre-Gaussian , and Airy-Ince-Gaussian . However, the chirped Airy Gaussian vortex (CAiGV) light bullets have not been explored, and the light bullets in the quadratic index medium have never been reported to the best of our knowledge.
The quadratic index, as a widely used medium, is a good example to implement graded index waveguides, optical fibers, and lenses [10, 26–29]. Based on the etching of one-dimensional sub wavelength gratings into a high-index slab waveguide to achieve the desired effective index distribution, a novel configuration for the implementation of sub wavelength-based graded-index devices was proposed . In the uniform medium, the distribution of the scattering force is proportional to the intensity distribution while the gradient force component is given in terms of an electric field amplitude , however, what will the radiation force happen in the non-uniform refractive index medium? It will be significant to investigate the 3D localized CAiGV light bullets and the spatial radiation force of the CAiGV beams in the quadratic index medium. The research of Airy related wave packet has become quite involved in applications [31–37] such as plasma generation and optical tweezing, almost all applications have been focused on cases without initial velocity. For the first time, we have demonstrated the 3D localized CAiGV wave packets carry the initial velocity to investigate the propagation properties of the light bullets in the quadratic index medium.
The paper is organized as follows. Firstly, we start from the model with paraxial differential equation and obtain the solutions of 3D localized CAiGV wave packets in section 2. Then in section 3, propagation properties of CAiGV light bullets are studied. Radiation force produced by spatial CAiGV wave packets is reported in section 4. Finally, we conclude the paper in section 5.
2. Theory of 3D CAiGV optical light bullets
We consider a (3+1)D dispersive and diffractive optical paraxial system in the quadratic index medium whose refractive index varies radially as n(r) = n0(1−α′2r2/2), α′ is a measure of the parabolic dependence (rate of decrease). In the paraxial approximation and in the absence of nonlinearity, the spatiotemporal propagation of the wave packet satisfies the (3 + 1)D paraxial differential equationEq. (1) can be rewritten as
As both the spatial and temporal coordinates can be readily rescaled in this expression, we here assume that dispersion and diffraction have the same quantitative effect along the propagation (i.e. Ldif f = Ldisp) . We choose α = Ldif fα′, therefore, the spatiotemporal evolution of an optical bullet can be simplified, Eq. (2) can be expressed as
In order to obtain solutions of Eq. (3) in the quadratic index medium, we consider a separation of variables and set a function of the form ϕ (X,Y,T,Z) = ϕ (X,Y,Z)A(T,Z), then we have the following equations
From Eq. (4), it is not difficult to obtain the finite-energy chirped Airy distribution with an initial velocity as the form , where Ai(·) is the Airy function, σ determines the direction of the Airy function envelope, and we choose σ = −1, an Airy distribution can be launched with its tails at the front and undergoes negative accelerating (i.e. decelerating) . at (0 < at ⩽ 1) is the decay factor in T direction, which enables a physical realization of the beam with finite energy [2, 3], ct is the initial velocity in T direction, which can be determined by the incident angles and directions , βt is the temporal chirp parameter . Under such initial conditions, the solution of Eq. (3) can be obtained as36], the solution of Eq. (5) can be constructed by using Eq. (7) and Eq. (5). After some algebraic calculations, the solution can be expressed as
Without loss of generality, we consider the first and second order vortex wave packets with m = 1 and m = 2, as
3. Analysis and discussion of CAiGV wave packets
Based on the exact analytical solutions in section 2, we can analyze the propagation properties of CAiGV light bullets in a quadratic index medium with α = 1.
In Eq. (6), it is easy to discover the singularity of b when Z = −1/(2βt), if the chirp parameter βt is negative, b will change to a negative one after this position, the direction of the chirp Airy function envelope will switch from its tails at the front to its main lobe at the front. On the other hand, if the chirp parameter βt is positive, it cannot change the direction of the chirp Airy function envelope. According to Eq. (6), we can show the evolution of such chirped Airy pulses during propagation in the temporal domain with different temporal chirp parameters in Fig. (1). As we can see in Figs. 1(b) and 1(e), the Airy pulse accelerates in the negative direction with the propagation distance increasing, the direction is opposite to the positive coordinate, which is the inverse of the usually displayed figures of positive accelerating Airy pulses with σ = 1 . As Fig. 1(a) shows, when the chirp parameter βt = −1, after reaches a focal point, the direction of the Airy function envelope in Z = 1 is opposite to that of the initial position Z = 0, but maintains negative accelerating. As Fig. 1(d) shows, it undergoes compression in the first propagation distance; then the pulse pattern breaks up with further increasing of the propagation distance; finally, a new Airy pattern with the rotational symmetry distribution is regenerated. In Figs. 1(c) and 1(f), the Airy pulse disperses during the propagation, the direction is the same to that of the positive coordinate.
In Eq. (8), when csc(αZ) = 0, βs = 0, there are special positions related to the potential width parameter in Z = (2l +1)L/4 with the oscillating period L = 2π/α and l is an integer, which will bring periodically inversion in spatial domain. Here we show the particular snapshots describing the evolution of the spatiotemporal second order CAiGV light bullets through the quadratic index medium in Fig. 2. The vortices in main lobe of light bullets rotate with propagation. The spatial directions of Figs. 2(b) and 2(c) are opposite due to the spatial direction switches in Z = π/2. The second order CAiGV light bullets propagate half period to the opposite direction when Z = π in Fig. 2(d), and another half period including Figs. 2(d)–2(f), there is no doubt that in Z = 2π it will have the same direction as the initial direction in Fig. 2(a).
We show the influence of the positive chirp parameter in propagation properties of the second order CAiGV light bullets through the quadratic index medium in Fig. 3. The light bullets accelerating in positive temporal domain when βt is positive in Figs. 3(a) and 3(d). Figures 3(b) and 3(e) are analytical results related to spatial part of Figs. 3(a) and 3(d), while Figs. 3(c) and 3(f) are corresponding numerical simulations. By comparing Figs. 3(b) with 3(c) and Figs. 3(e) with 3(f), there is very good qualitative agreement between analytical results and numerical simulation of the spatial part of the second order CAiGV light bullets.
One can notice that the CAiGV light bullets with the negative temporal chirp parameter βt = −1 in Fig. 4 reverse in temporal domain, which means the light bullets changing to main lobe at the front comparing with its tails at the front in Figs. 2 and 3. This matches well with our discussion in Fig. 1, and as we reported, the reverse position can be controlled by selecting different negative temporal chirp parameters. With different temporal chirp parameters, the temporal part of CAiGV light bullets may have big changes, but they still keep the same vortices rotating characteristic and spatial direction switch by comparing Fig. 4 with Figs. 2 and 3. The rotating propagation trajectory of other CAiGV light bullets with different topological charges in the quadratic index medium is similar to that of the first and second order CAiGV light bullets except for number of vortices in main lobe of light bullets, and different side lobes appear though with the same parameters as those in Fig. 4.
In order to discover the effect of the initial velocity, we plot the beam center . In Figs. 5(a)–5(c), the CAiGV distributions undergo parabolic trajectory, which tends to Airy vortex distribution for smaller χ0 while tends to Gaussian vortex distribution for bigger χ0. The beam center changes vividly with the different initial velocities, the positive initial velocity brings accelerating while the negative initial velocity brings decelerating. Though the beam center in spatial domain undergoes quasi parabolic profile, the distribution factor changes their bends characteristics and the initial velocity changes their accelerating properties. Figures 5(d)–5(f) are CAiGV light bullets controlled by the initial velocity in X,Y, and T directions. We can see that the negative temporal initial velocity has the same positive accelerating effect as the positive spatial initial velocity in Fig. 5(d), in other words, the opposite initial velocities have same effects in spatial and temporal domains due to opposite σ. The features of CAiGV light bullets propagation seen in Fig. 5 can be understood qualitatively as follows. In dimensionless coordinates, we take T direction for example, from Eq. (6) one can conclude that this distribution follows a ballistic trajectory  in the T–Z plane, which is described by T = −Z2/4 − ctZ when σ = −1, βt = 0.dT/dZ = gZ – ct, d2T/dZ2 = −1/2 = g, where g plays the role of negative acceleration. We notice here when the initial velocity is negative ct < 0, the distribution will initially propagate in the positive direction until it stalls at Z = −2ct, at this point due to negative acceleration, the maximum deflection is . That is, the pulse will undergo positive propagation and then negative accelerating if the initial velocity is in the same direction to the negative acceleration. When σ = 1, the ballistic trajectory is described by T′ = Z2/4 + ctZ, dT′/dZ = g′Z + ct, d2T′/dZ2 = 1/2 + g′, where g′ plays the role of positive acceleration. In spatial domain with opposite σ, the beam will undergo negative propagation until it stops at Z = −2ct, and then positive accelerating if the initial velocity is negative ct < 0. From what has been discussed above, the same initial velocity has the opposite effect in spatial (σ = 1) and temporal (σ = −1) domains.
Will the spatial chirp parameter influence the spatial reverse properties? We show the propagation properties of the first order CAiGV light bullets through the quadratic index medium with the positive spatial chirp parameter in Figs. 6(a)–6(c) and the negative spatial chirp parameter in Figs. 6(d)–6(f). One can easily notice different spatial switches in Figs. 6(a) and 6(b) (βs = 0.5), Figs. 6(d) and 6(e) (βs = −0.5) by comparing with Figs. 4(d) and 4(e) (βs = 0). The features of CAiGV light bullets propagation seen in Fig. 6 can be understood qualitatively as follows. From Eq. (8), it is easy to discover the special positions in Z = arctan[−α/(2βs)]/α +lL/2, which are affected by the spatial chirp parameter βs. After the position Z = arctan[−α/(2βs)]/α + lL/2, the spatial direction will change to a negative one, namely, the direction of the chirp Airy function envelope will switch from its tails at the front to its main lobe at the front. When βs → 0, the reverse position tends to Z = (2l + 1)L/4. The oscillating period L = 2π/α of CAiGV light bullets in a quadratic index medium can not be influenced by the chirp parameter by comparing Figs. 6(c) and 6(f) with Fig. 4(f).
We show the propagation of the first order spatial CAiGV wave packets with different spatial chirp parameters and initial velocities in Fig. 7. It is obviously that the inverse positions in Figs. 7(a) and 7(b) move ahead of the initial reverse position Z = π/2 (red dotted lines), which is caused by the positive spatial chirp parameters with l = 1. And the inverse positions in Figs. 7(b) and 7(c) move behind of the initial reverse position, which are caused by the negative spatial chirp parameters with l = 0. The difference between l comes from the negative spatial chirp parameter which can bring a reverse position, similar to the temporal domain we discussed. Initial velocities bring accelerating in Fig. 7(b) and decelerating in Fig. 7(d), but do not influence the reverse positions with the same spatial chirp parameters. This matches well with our theoretical derivation.
4. Radiation force produced by spatial CAiGV wave packets
Next, we study the radiation force of CAiGV distributions on a Rayleigh dielectric particle, the particle radius r0 is sufficiently small comparing with the wavelength (i.e. r0 ≤ λ/20) and can be seen as a point dipole in the light field. When the particle is in steady state, the scattering force and the gradient force are given by [30, 38]
In the uniform medium, the distribution of the scattering force is proportional to the intensity distribution, while the gradient force component is given in terms of an electric field amplitude . However, what will happen in the non-uniform refractive index medium?
We plot the distributions of the intensity, scattering force, and transverse gradient force of the first order spatial CAiGV distributions for the particles at different positions through the quadratic index medium with the initial velocity cx = cy = 3 and the chirp parameter βs = 0.5 in Fig. 8. The initial velocities have the same trajectory effect in intensity and radiation force. The propagation trajectory of Figs. 8(a1)–8(f1) is show in Fig. 7(b), one can easily see that the radiation forces Figs. 8(a2)–8(f2) and 8(a3)–8(f3) still maintain the periodically oscillation, reversion and recovery properties because of the quadratic index potential. However, due to the medium refractive index varies radially, the main lobe of the scattering force of the CAiGV beams in Figs. 8(a2) and 8(d2) and transverse gradient force in Figs. 8(a3) and 8(d3) disappear, which are not similar to the intensity distribution in Figs. 8(a1) and 8(d1). With the initial velocity considered, only with Z = L/2 in Figs. 8(d2) and 8(d3) the center position keeps the same, the radiation force in main lobe appears after passing the center position in Figs. 8(b2), 8(c2), 8(e2) and 8(f2), Figs. 8(b3), 8(c3), 8(e3) and 8(f3), which means the radiation force transfer to the main lobe and gradually comes back into side lobe in a half period Z = (2l + 1)L/2 and will recover in a complete period Z = lL. Owing to the side lobes from the first order CAiGV distribution, more traps will be generated along the propagation of the CAiGV distribution comparing with traditional Gaussian beams. The chirp parameter in CAiGV beams can change the reverse positions of radiation forces, while other beams such as Airy Gaussian beams in a similar setup can not. By changing the initial velocity and the chirp parameter, the radiation force we study here will obtain different trajectories and may have significant sense in optical tweezing and bio-medical field.
The exact solution of the (3 + 1)D paraxial differential equation in the quadratic index medium is obtained in Cartesian coordinates by using the method of separation of variables. For the first time, we have demonstrated the 3D localized CAiGV wave packets to investigate the propagation properties of the light bullets in the quadratic index medium. We have presented some typical examples of the obtained solutions on the basis of parameters: the temporal chirp parameter βt, the spatial chirp parameter βs, the initial velocity cs, the distribution factor χ0, and the topological charge m. The results have shown that the wave packets can retain their periodically reverse in spatial domain during propagation. The negative temporal chirp parameter can bring temporal reverse, while the spatial chirp parameter can affect the reverse position of a quadratic index medium. The beam center changes vividly with the different initial velocities, the positive initial velocity can bring accelerating while the negative initial velocity can bring decelerating firstly and then accelerating. The same initial velocity has opposite effect in spatial domain (with σ = 1) and temporal domain (with σ = −1). The beam center and propagation trajectory in spatial domain undergoes quasi parabolic profile, the distribution factor changes their bends characteristics and the initial velocity changes their accelerating properties. We also find that the optical forces have the periodically reversion and recovery because of the quadratic index potential. What we reported here can obtain new radiation force trajectory in Rayleigh dielectric particles and may have significant sense in optical tweezing and bio-medical field [30, 38].
Our method is also possible in generation of other kinds of light bullets in quadratic index medium. And our demonstration may lead to potential particle manipulation, signal processing applications [7, 8] of such wave packets in quadratic index medium with its positive accelerating and negative accelerating, reversible and vortex properties.
National Natural Science Foundation of China (11374108, 11374107); CAS Key Laboratory of Geospace Environment, University of Science and Technology of China; Innovation Project of Graduate School of South China Normal University (2016lkxm18).
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