## Abstract

The evolution of the three-dimensional (3D) self-accelerating Airy-Ince-Gaussian (AiIG) and Airy-Helical-Ince-Gaussian (AiHIG) light bullets is investigated by solving the (3+1)D linear spatiotemporal evolution equation of an optical field analytically. As far as we know, the numerical experimental demonstrations of the Ince-Gaussian (IG) and Helical-Ince-Gaussian (HIG) beams in various modes are first developed to study the evolution characteristics of the different 3D spatiotemporal light bullets. A conclusion can be drawn that the different photoelastics, pulse stacked, boundary, elliptical ring and physically separated in-line vortices can be achieved by adjusting the ellipticity, the evolution distance and the mode-number of light bullets.

© 2016 Optical Society of America

## 1. Introduction

In recent years, the self-accelerating optical beams have been one of the most exciting research fields. As a typical example, the Airy beams, which have the characteristics of self-accelerating [1–8], nondiffracting, and self-healing [9, 10], have achieved extensive attention over the past decade. This intriguing wave packet has been predicted for the first time by Berry and Balazs in 1979 and demonstrated theoretically and experimentally by Christodoulides et al. in 2007 [1,11].

Another hot topic, the light bullet, was first proposed in an early comprehensive review [12]. While a recent overview has been reported by Mihalache in [13]. So far, more and more researchers have paid great attention to the 3D spatiotemporal light bullets. For instance, Valtna-Lukner et al. have measured the spatiotemporal field of ultrashort pulses with complex spatiotemporal profiles [14]. Subsequently, D. Abdollahpour et al. have demonstrated the realization of spatiotemporal Airy light bullets by combining a spatial Airy beam with an Airy pulse in time [15]. Zhong et al. have also done some research on light bullets [16–19]. For example, they have studied the three-dimensional finite-energy Airy self-accelerating parabolic-cylinder light bullets, the accelerating Airy-Gauss-Kummer localized wave packets, the three-dimensional localized Airy-Laguerre-Gaussian wave packets and the self-decelerating Airy-Bessel light bullets. In addition, Airy-Hermite-Gaussian (AiHG) and Airy-Laguerre-Gaussian (AiLG) wave packets have been studied by F. Deng [20] and W. P. Zhong [18], respectively. However, there is few paper investigating on the Airy-Ince-Gaussian (AiIG) light bullets.

The Ince-Gaussian (IG) modes [21–27], which constitute the third complete family of transverse eigenmodes of stable resonators. The IG modes are exact and orthogonal solutions of a paraxial wave equation in elliptic coordinates and can be considered continuous transition modes between Hermite Gaussian (HG) modes and Laguerre Gaussian (LG) modes. The transverse distribution of IG modes is described by Ince polynomials. Most of the evoluting and resonating characteristics of HG modes and LG modes can be extended to IG modes. The IG modes, HG modes [28, 29] and LG modes [30] together constitute the complete solution set of the paraxial wave equation. When the ellipticity approaches to zero and infinity, the IG modes appulsively change into the HG modes and LG modes, respectively. An extremely attractive extension in 3D is to form the AiIG light bullets and Airy-Helical-Ince-Gaussian (AiHIG) light bullets by combining the 1D Airy pulse with the 2D IG beam and Helical-Ince-Gaussian (HIG) beams, respectively.

In this paper, we demonstrate the numerical experiments of the IG and HIG beams and analyze the evolution characteristics of the self-accelerating 3D spatiotemporal AiIG and AiHIG light bullets. The results show that the intensity distributions of these light bullets are influenced by the ellipticity, the evolution distance and the mode number.

The structure of this paper is as follows. In section 2, the model with (3+1)D linear spatiotemporal evolution equation of an optical field is formulated and the solutions of the self-accelerating AiIG and AiHIG light bullets are shown. Then, we demonstrate the numerical experiment of the IG beams, the HIG beams and the different profiles of the 3D light bullets in section 3. Finally, the paper is concluded in section 4.

## 2. The models of the AiIG and AiHIG light bullets

The IG beams and HIG beams in combination with other nondiffracting field configurations can be used to describe (3+1)-dimensional finite energy wave packets in the presence of diffraction and dispersion. In such a case, the wave packet in the spatiotemporal domain obeys [1, 31]:

*X,Y,Z,T*) denotes the complex envelop of the optical field.

*X*=

*x/w*

_{0},

*Y*=

*y/w*

_{0}mean the dimensionless transverse coordinate, and

*T*=

*t/t*

_{0}is the dimensionless time, here

*w*

_{0}and

*t*

_{0}are the spatial scaling parameter and the temporal scaling parameter, respectively. $Z=\frac{z}{k{w}_{0}^{2}/2}$ represents the normalized evolution distance, in which $k{w}_{0}^{2}/2$ is the Rayleigh length,

*k*= 2

*π/λ*

_{0}is the wave number,

*λ*

_{0}is the vacuum wavelength.

In Eq. (1), assuming:

where Ψ*(*

_{G}*X,Y,Z*) = 1

*/w*(

*Z*)exp(−(

*X*

^{2}+

*Y*

^{2})

*/w*

^{2}(

*Z*) +

*iZ*(

*X*

^{2}+

*Y*

^{2})

*/w*

^{2}(

*Z*) −

*i*arctan

*Z*) and $w(Z)=\sqrt{1+{Z}^{2}}$. In the transverse plane perpendicular to

*Z*, the elliptic coordinate transformation is defined as:

*X*=

*f*(

*Z*)

*coshξ cosJ*,

*Y*=

*f*(

*Z*)

*sinhξ sinJ*and

*Z*=

*Z*,

*ξ*≥ 0 and 0 ≤

*J*< 2

*π*are the radial and angular elliptic variables. And

*f*(

*Z*) =

*f*

_{0}

*w*(

*Z*)

*/w*

_{0}, where

*f*

_{0}is the semi-focal separation at waist plane

*Z*= 0. Where

*M*(

*ξ*),

*N*(

*J*) and

*Z*

_{1}(

*Z*) are real functions. Substituting Eq. (2) into Eq. (1), four differential equations are obtained:

*p*and

*a*are the separation constants, and $\epsilon =2{f}_{0}^{2}/{w}_{0}^{2}$ is the ellipticity parameter. The ellipticity

*ε*, the waist spot

*w*

_{0}, and the semi-focal separation

*f*

_{0}are the physically important parameters for describing the transverse structure of the IG modes. The ellipticity parameter

*ε*adjusts the ellipticity of the mode, while the parameters

*w*

_{0}and

*f*

_{0}scale its physical size [22,23].

Equation (3) is solved by using the Fourier-transform method. The initial unchirped Airy pulse is *A*(*Z* = 0,*T*) = *Ai*(*σ T*)exp(*α σ T*) [1], where *α* (0 < *α* < 1) is the decay parameter and *Ai*(·) is the Airy function. As introduced above, we focus on discussing the solution with *σ* = 1 (*σ* = −1 represents the self-decelerating, *σ* = 1 means the self-accelerating). The intriguing solution is obtained:

*s*instead of

*T*.

The solution of Eq. (4) can be expressed in the form of

Equation (6) is the Ince equation [21, 24, 32], which is an exception of the Hill equation. If *iξ* = *J*, Eq. (5) will change into Eq. (6), vice versa. The solutions of Eq. (6) are obviously the even and odd Ince polynomials of order *p* and degree *m*, generally denoted as
${C}_{p}^{m}(J,\epsilon )$ and
${S}_{p}^{m}(J,\epsilon )$, where 0 ≤ *m* ≤ *p* for an even function, 1 ≤ *m* ≤ *p* for an odd function, and parameters (*p,m*) have the same parity [21, 24, 25, 32], namely, (−1)^{p}^{−}* ^{m}* = 1. To achieve the solutions, the functions of the same parity in both

*J*and

*ξ*should be provided, which meet continuity in the whole space; the even and odd IG beams can be expressed as:

*D*and

_{e}*D*are dimensionless parameters,

_{o}*δ*= 1/(

*kw*

_{0}), the superscripts

*e*and

*o*describe even and odd modes, respectively. When (

*p,m*) = (0,0), the IG beams are obviously the lowest-order Gaussian beams. The relationship of IG beams, HG beams and LG beams will be analysed as follows [21–27]. When the ellipticity

*ε*→ 0, the transition from $I{G}_{pm}^{e,o}$ to $L{G}_{nl}^{e,o}$ occurs, where LG stands for the LG function. Simultaneously, the indices of both modes are related as:

*l*=

*m*and

*n*= (

*p*−

*m*)/2 in the limit. When

*ε*→ ∞, the transition from $I{G}_{pm}^{e,o}$ to $H{G}_{{l}_{X}{l}_{Y}}$ occurs, where HG stands for the HG function. In the limit, the indices are related as: for even IG beams

*l*=

_{x}*m*and

*l*=

_{y}*p*−

*m*, on the contrary, for odd IG beams

*l*=

_{x}*m*− 1 and

*l*=

_{y}*p*−

*m*+ 1.

The 3D AiIG light bullets can be achieved by combining the 1D self-accelerating finite-energy Airy wave function and the 2D IG functions in elliptic coordinate. Substituting the Eqs. (7)–(10) into the Eq. (2), the solution of the Eq. (1) is obtained. The self-accelerating even and odd AiIG light bullets can be expressed as:

The HIG beams can be constructed by forming a linear combination of the even and odd IG beams [21–27]. Similarly, the self-accelerating AiHIG light bullets can also be taken as a linear combination of the self-accelerating even and odd AiIG light bullets. Consequently, the HIG beams and the AiHIG light bullets can be denoted respectively as:

*m*> 0 because ${\mathrm{\Psi}}_{+}^{o}(X,Y,Z,T)$ is not defined as

*m*= 0. The sign ’+’ means the positive helicities. Here, these HIG beams are described by

*N*= 1 + (

*p*−

*m*)/2 elliptic rings and have

*m*physically separated in-line vortices, each with unitary topological charge such that the total charge (along a closed trajectory enclosing all the vortices) is

*m*; thus, a single elliptic donut can be obtained with modes of

*p*=

*m*. The ellipticity of the rings is controlled by the ellipticity parameter. The changes in ellipticity are more dramatic for larger values of

*p*and

*m*than for smaller values. As the ellipticity approaches to zero, these separated vortices spatially collapse to a single vortex with the total charge of

*m*, i.e., the HIG beams tend toward the Laguerre-Gaussian beams [21–24, 26, 27].

## 3. Discussion of the solutions

In order to investigate the evolution properties of the accelerating 3D spatiotemporal wave packets, we have calculated the accurate analytical solutions, i.e. Eqs. (11), (12) and (14). Apparently, the varieties of the even and odd AiIG wave packets and the AiHIG wave packets can be constructed by adjusting the parameters (*p,m*). Therefore, the evolution properties of the light bullets are investigated by employing two typical cases of *p* = *m* and *p* ≠ *m*. In these two cases, we study the different situations of the light bullets with varying parameters and discover many exciting phenomenons.

#### 3.1. Case p = m

For simplicity, we take the parameters *p* = *m* = 3. As far as we know, this is the first time to demonstrate the numerical experiment of the IG and HIG beams for studying the evolution properties of the AiIG and AiHIG light bullets, as shown in Figs. 1–3. As revealed in Figs. 1–3, the initial beam patterns vary with the ellipticity *ε*. The Figs. 1(a1)–1(c1), 2(a1)–2(c1) and 3(a1)–3(c1) are the intensity patterns of the initial input of even IG beams, odd IG beams and HIG beams, respectively. The Figs. 1(a2)–1(c2), Figs. 2(a2)–2(c2) and 3(a2)–3(c2) are the phase patterns of the initial input of even IG beams, odd IG beams and HIG beams, respectively. The holograms are obtained by calculating the off-axis interference patterns between the complex amplitude profiles of IG beams and HIG beams at the *Z* = 0 plane and a plane wave [see Figs. 1(a3)–1(c3), 2(a3)–2(c3) and 3(a3)–3(c3)]. For the numerical experimental generation, an initial beam has been launched to reconstruct the off-axis computer-generated holograms [see Figs. 1(a4)–1(c4), 2(a4)–2(c4) and 3(a4)–3(c4)] of the desired beam profiles [33]. The transverse intensity patterns are obtained at the input plane [see Figs. 1(a5)–1(c5), 2(a5)–2(c5) and 3(a5)–3(c5)].

Fortunately, we have achieved the accurately numerical experimental results in accordance well with analytical results. Due to the relation among IG beams, LG beams and HG beams, we can change the parameter *ε* to analyze the IG beams and HIG beams. In this paper, when the ellipticity *ε* → 0,
$I{G}_{33}^{e,o}$ can be changed into
$L{G}_{03}^{e,o}$; when the ellipticity *ε* → ∞,
$I{G}_{33}^{e,o}$ can be turned into *HG*_{30} and *HG*_{21}. When *p* = *m* = 3, the HIG beams can be described by a single elliptical ring and three coaxial vortexes. Consequently, a single elliptical ring is obtained, as shown in Figs. 3(b3) and 3(b5). As revealed in Fig. 3, the number of coaxial vortexes varies with the values of the ellipticity *ε*. When the ellipticity *ε* → 0, these separated vortexes spatially collapse to a single vortex with the total charge of *m*. When the ellipticity *ε* = 2 or *ε* → ∞, the three coaxial vortexes turn up. However, when *ε* = 2, it is HIG beams with m coaxial vortexes; when *ε* → ∞, it is the HHG beams, and the number of the vortexes is decided by the number of the HHG beams at the point where the real and imaginary parts are zero simultaneously.

Figures. 4–6 are the intensity distributions of the self-accelerating light bullets of even AiIG, odd AiIG and AiHIG, respectively. We investigate the influence of the normalized evolution distance *Z* and the ellipticity *ε* to the evolution of the light bullets. Compared the Figs. 4–6 to Figs. 1–3, the transverse intensity of the light bullets is accordance well with their numerical experiment intensity for different ellipticity. With the increasing of the evolution distance of the light bullets, the profiles of the lobes are more and more similar to the real bullets. It is amazing that they will become photoelastics, and have different shapes. The pulse stacked light bullets are decreased with the evolution distance of the light bullets increasing, especially near *x* = 0. It is noted that the light bullets are accelerated in the temporal domain. What is more, the each lobe of the light bullets is equably extending along the x-axis and y-axis with the increasing of the evolution distance.

#### 3.2. Case p ≠ m

For a deeper research, we have investigated the case with *p* ≠ *m*. The intensity distributions of the even AiIG light bullets with (2,0) and (3,1) modes by varying the ellipticity is shown in Fig. 7, while Fig. 8 gives that of the odd AiIG light bullets with (3,1) and (4,2) modes at different *Z* plane. Thus, we have obtained a better understanding of the light bullets with different p and m in a relatively comprehensive way. In addition, the corresponding IG beams are analyzed with numerical experiment, and the results meet the expection, as shown in Figs. 7–9. The results show that either odd or even modes, the evolution cross section of the light bullets reaches to an ellipse with the increasing of the mode numbers. Actually, the number of the boundaries and lobes of the light bullets increases with the increasing of the mode numbers. The boundaries of the odd light bullets on x-axis are less than that of the even light bullets under the same mode numbers, because the nodal line at the *ξ* = 0 is not taken into account for the odd mode. With the increasing of the evolution distance, the pulse stacked is decreasing while the profiles of the lobes become photoelastics, as shown in Figs. 8(b1)–8(b2). The scale of the high order light bullets is larger than the low order one in the x-y plane.

For simplicity, the case *p* = 3, *m* = 1 has been acted as an example of the *p* ≠ *m* to study the properties of the self-accelerating AiHIG light bullets and verify the numerical experiment for the HIG beams. As revealed in Fig. 10(a2), when *ε* = 2, there are two elliptical rings and a physically separated in-line vortex on the transverse of the AiHIG light bullets, which is accordance well with the composition of the HIG beams in above. When *ε* → 0, the profiles of the AiHIG light bullets with (3,3) and (3,1) modes are similar to the elliptic cylinder, as shown in Fig. 6(a1) and Fig. 10(a1). As the ellipticity approaches to zero, the HIG beams tend toward the LG vortex beams. In the Fig. 10, when *m* = 1, the coaxial vortexes of the AiHIG light bullets are along the Y-axis, which is highly accordance with the direction of the phase singularity. For the high order AiHIG light bullets, for instance *m* = 3, the coaxial vortexes of the AiHIG light bullets are along the X-axis. Similarly, the beam widths of the initial AiHIG light bullets are less than 2 *f* under these two modes.

## 4. Conclusion

To summarize, we have demonstrated the existence of the 3D spatiotemporal AiIG and AiHIG light bullets, which are governed by the linear spatiotemporal evolution equation of an optical field. The AiHIG light bullets can be constructed with a linear combination of the even and odd AiIG light bullets. For the first time, we have demonstrated the numerical experiment of the IG and HIG beams to investigate the evolution properties of the AiIG and AiHIG light bullets. The 3D spatiotemporal self-accelerating AiIG and AiHIG light bullets have been studied in the cases of *p* = *m* and *p* ≠ *m*. The results show that the photoelastics, the pulse stacked, the boundaries, the elliptical rings and the physically separated in-line vortices can be controlled by adjusting the ellipticity, the evolution distance and the mode numbers. In addition, the AiIG and AiHIG light bullets have constituted the continuous and accurate transition modes between AiHG and AiHHG light bullets and AiLG (even and odd) and AiLG light bullets. And the proposed 3D light bullets in our work may bring potential applications in optical communication and biological applications [15].

## Funding

National Natural Science Foundation of China (NSFC) (11374108, 10904041, 11374107); Foundation of Cultivating Outstanding Young Scholars (“Thousand, Hundred, Ten” Program) of Guangdong Province in China; CAS Key Laboratory of Geospace Environment, University of Science and Technology of China.

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