Self-accelerating Airy-Laguerre-Gaussian light bullets in a two-dimensional strongly nonlocal nonlinear medium

Zhenkun Wu; Zhiping Wang; Hao Guo; Wei Wang; Yuzong Gu

doi:10.1364/OE.25.030468

1. Introduction

The self-accelerating beams related to the Airy or Bessel functions have attracted a considerable attention in the past few decades. As one solution of the potential-free Schrödinger equation, Airy function was first theoretically discovered by Berry and Balazs in 1979 [1]. Later, in 2007 the paraxial accelerating Airy beam was experimentally observed by Siviloglou and Christodoulides [2, 3]. In experiment, the common method for generating Airy beam is to impose a cubic phase to one Gaussian beam using spatial light modulator and subsequently apply an optical lens for Fourier transformation [2, 3]. It was demonstrated that Airy beams can exhibit remarkable and interesting properties: self-acceleration, self-healing, and nondiffraction over many Rayleigh lengths [2, 3]. And such one–and two–dimensional (1D and 2D) optical beams were extensively studied in optically linear medium, nonlinear dielectrics [4], photonic crystals [5], Bose–Einstein condensates [6], in atomic vapor with electromagnetically induced transparency [7], chiral media [8] and on the surface of a gold metal film [9] or on the surface of silver [10]. The propagation of Airy beam in the strongly nonlocal nonlinear medium (SNNM) has potential application in optical micromanpulation and optical switching [11]. Shen et al. have studied the anomalous interaction of Airy beams in nonlocal nonlinear media [12], in which nonlocal nonlinearity also affects the interaction of out-of-phase bright solitons and dark solitons. The Zhou et al have obtained the analytical expression of an Airy beam in the SNNM and further investigated its propagation law [11]. It is shown that, owing to the SNNM, the normalized intensity distribution and the aforementioned parameters versus the axial propagation distance are all periodic. The first-order and the second-order Airy vortex beams follow a sinusoidal periodic trajectory when propagating through a SNNM have been presented by Zhu et al [13].

Recent interest in the study of the three–dimensional (3D) spatiotemporal wave packets or light bullet has been abundant and have undergone rapid development. The spatiotemporal Airy light bullets by combining an Airy pulse in time with two spatial Airy beams have been introduced by Abdollahpour et al [14]. Generally, such spatiotemporal wave packets can be obtained by Airy pulse in combination with other nondiffracting field configurations. For example, Chong et al. introduced the 3D Airy-Bessel light bullets [15] by combining an Airy pulse with a 2D Bessel beam. It is worth mentioning that an oblique spatitemporal Airy-Bessel wave packet was reported by Eichelkraut et al [16]. Zhong et al. [17] and Deng et al. [18] have individually studied the dynamics of the 3D linear Airy–Laguerre–Gaussian (AiLG) and Airy–Hermite–Gaussian (AiHG) beams in free space based on the polar coordinate and Cartesian coordinate. The 3D Airy–Kummer–Gaussian (AiKG) and Airy–Ince–Gaussian (AiIG) localized wave packets were also reported by Zhong et al. and Deng et al. in Refs [19]. and [20], respectively. Note that, such spatiotemporal wave packets can retain their energy features over several Rayleigh lengths in free space. However, if one launches an light bullet in 2D SNNM, the propagation is very different.

Quite surprisingly, such a problem has not been considered in the research literature. In this work, we report an exact 3D spatiotemporal nonlinear light bullets solution of (3 + 1)D nonlocal nonlinear Schrödinger equation (NNLSE), which is constructed by 1D self-accelerating finite-energy Airy pulse with 2D Laguerre–Gaussian beams in the polar coordinate. We find that the behaviour of Self-accelerating Airy–Laguerre–Gaussian light bullets can be considerably affected by the radial mode number, the azimuthal mode number and modulation depth. Besides, the propagation property of Airy–Laguerre–Gaussian light bullets, in 2D SNNM, is also investigated both analytically and numerically. Its profile undergo a profound change during propagation due to SNNM, which is quite different from the case in free space. Comparing with previous literature [15–20], the main novelties of our work can be summarized as the following key advantages: First and foremost is that we are interested in showing the spatiotemporal evolution equation of an optical field. Using the method of separation of variables, an exact solution in form of nonlinear self-accelerating Airy–Laguerre–Gaussian light bullet solution is reported. Second, in our work, the wave packet propagates in a NL medium and is an eigenmode of NLSE, which is quite different from the linear light bullets obtained in pioneering studies [17–20]. Third, the evolution of Airy–Laguerre–Gaussian light bullets based on various modes in SNNM is individually considered during propagation, which has attracted little attention to the best of our knowledge.

The structure of this paper is organized as: in Section2, we firstly introduce the theoretical model with the (3 + 1) D nonlinear spatiotemporal evolution equation of an optical field in the polar coordinate, secondly obtain the 1D Airy pulse and 2D Laguerre–Gaussian beams solution, respectively, and lastly construct the self-accelerating Airy–Laguerre–Gaussian light bullets; in Section3, we investigate the propagation properties of the light bullets mentioned in Section2 in detail, and meanwhile compare the results with those in free space; in section 4, we conclude the article.

2. Theoretical model

Anonlocal model is provides a radical simplification and allows for an elegant description of soliton collisions, interactions, and deformations in two and three dimensions. In the polar coordinates we begin our analysis by considering that spatiotemporal wave packet propagates in 2D nonlocal nonlinear medium, the evolution of which is governed by the dimensionless (3 + 1)D NNLSE [21, 22]:

i \frac{\partial ψ}{\partial Z} + \frac{1}{2} (\frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial φ^{2}} + \frac{\partial^{2}}{\partial T^{2}}) ψ + n [I (r, φ)] ψ = 0

with the 2D nonlinear perturbation of the refractive index

n [I (r, φ)]

being given by

n [I (r, φ)] = \int_{- \infty}^{+ \infty} \int_{0}^{2 π} R (r - r^{'}) {| ψ^{'} (r^{'}, φ^{'}, Z) |}^{2} r^{'} d r^{'} d φ^{'}

where

ψ (r, φ, T, Z)

represents the complex envelop of the optical field,

r = \sqrt{X^{2} + Y^{2}}

with

X = x / r_{0}

,

Y = y / r_{0}

, and

T = t / t_{0}

are normalized coordinates, and

r_{0}

is the initial beam width of Gaussian beam,

t_{0}

is the temporal scaling parameter,

Z= z / Z_{R}

is the normalized propagation distance, where

Z_{R} = k r_{0}^{2}

is Rayleigh length,

k = 2 π / λ

is the wave number (

λ = λ_{0}

, where

λ_{0}

is the vacuum wavelength).

R (r)

corresponds to the normalized symmetrical real spatial response function of the medium. In paper [12], response function

R (r) = \exp (- r^{2} / σ^{2}) / (\sqrt{π} σ)

is considered, where

σ

is the characteristic length of response, and it describes a local and a strongly nonlocal media when

σ \to 0

and

σ \to \infty

, respectively. In general, according to literature [21, 22], the realistic forms of the nonlocal response functions depend on the underlying physical process of the materials.

We consider a strong nonlocality in which the beam width is much smaller than the width of the response function, i.e., the characteristic length of the material. In this case, the response function $R (r)$ can be expanded in Taylor’s series. If the response function is expanded to the second order, the (3 + 1)D NNLSE can be deduced to the Snyder-Mitchell linear model [22–25]

i \frac{\partial ψ}{\partial Z} + \frac{1}{2} (\frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial φ^{2}} + \frac{\partial^{2}}{\partial T^{2}}) ψ - \frac{1}{2} γ^{2} P_{0} r^{2} ψ = 0

where

γ

is a material constant associated with the response function, and

P_{0} = {\iint | ψ (r, φ) |}^{2} r d r d φ

is the input power of the beam. In the following numerical example, the relevant parameters of strong nonlocality are considered as

γ = \sqrt{2}

and

P_{0} = 1

for convenience without special statement [24]. To find the solution of Eq. (3), we assume a solution of the form

ψ (Z, r, φ, T) = M (Z, r, φ) N (Z, T)

, and substitute

ψ (Z, R, φ, T)

into Eq. (3). Using the separation of variables method, one ends up with the following two partial differential equations:

i \frac{\partial N (Z, T)}{\partial Z} + \frac{1}{2} (\frac{\partial^{2}}{\partial T^{2}}) N (Z, T) = 0

i \frac{\partial M (Z, r, φ)}{\partial Z} + \frac{1}{2} (\frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial φ^{2}}) M (Z, r, φ) - r^{2} M (Z, r, φ) = 0

We first focus on Eq. (4a) and specifically investigate the dynamics of finite-energy Airy pulse in the temporal domain $T$ , by considering a specific input into the system (at $Z=0$ ) of the form $N (Z = 0, T) = A i (σ T) \exp (σ a T)$ , where $σ = \pm 1$ ( $σ = 1$ is relevant to the self-accelerating, and $σ = - 1$ is corresponding to the self-decelerating), the Airy function is defined as $A i (T) = \frac{1}{2 π i} \int_{- i \infty}^{+ i \infty} \exp (T t - t^{3} / 3) d t$ and $a$ ( $0 < a < 1$ ) is the decay parameter [2,3]. Under such an initial condition, we can solve the Eq. (3) directly with the self-accelerating and the self-decelerating Airy pulses for $σ = \pm 1$ , the solutions are obtained

N_{+} (Z, T) = A i (T - \frac{Z^{2}}{4} + i a Z) e^{a T - \frac{1}{4} a Z^{2} + i (Z - \frac{1}{24} Z^{3} + \frac{1}{2} a^{2} Z + \frac{1}{2} T Z)}

N_{-} (Z, T) = A i (- T - \frac{Z^{2}}{4} + i a Z) e^{- a T - \frac{1}{4} a Z^{2} + i (Z - \frac{1}{24} Z^{3} + \frac{1}{2} a^{2} Z - \frac{1}{2} T Z)}

The intensity profiles at various distances for Airy pulse $N_{+} (Z, T)$ ( $N_{-} (Z, T)$ ) are displayed in Fig. 1(a1) and (a2) (Fig. 1(b1) and (b2)), respectively. In particular, for $σ = 1$ ( $σ = - 1$ ), as shown in Fig. 1(a2) (Fig. 1(b2)), the pulse is accelerating with propagation distance in the positive (negative) $T$ direction, as indicated by the black arrows. One can note that upon propagation, the individual pulses attenuate slowly and acquire a long flat tails. In order to compare our results with anterior literature [2, 3, 26, 27], we focus on discussing the self-accelerating Airy pulse solution (5a) with $σ = 1$ in this paper.

Fig. 1 (a1) and (b1) Contour-plots of accelerating finite-energy Airy pulse intensity distribution, as functions of the propagation distance with $σ = + 1$ and $σ = - 1$ , respectively. (a2) and (b2) Intensity profiles of the pulse along the positive and negative $T$ direction corresponding to (a1) and (b1). The parameters are $r_{0} = 100 μ m$ , $λ = 530 n m$ and $a = 0.1$ . Not that $Z = 5$ when $z = 59 c m$ .

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Now, we solve Eq. (4b) using the self-similar method. Following [28,29], we define the complex field as $M (Z, r, φ) = A (Z, r, φ) e^{i B (Z, r)}$ , where $A (Z, r, φ)$ and $B (Z, r)$ are real function. Substituting $M (Z, r, φ)$ into Eq. (4b), we find the following two coupled equations for the amplitude $A (Z, r, φ)$ and the phase $B (Z, r)$ :

- \frac{\partial B}{\partial Z} + \frac{1}{2} (\frac{1}{A} \frac{\partial^{2} A}{\partial r^{2}} + {(\frac{\partial B}{\partial r})}^{2} + \frac{1}{r A} \frac{\partial A}{\partial r} + \frac{1}{r^{2} A} \frac{\partial^{2} A}{\partial φ^{2}}) - r^{2} = 0

\frac{1}{A} \frac{\partial A}{\partial Z} + \frac{1}{2} (\frac{2}{A} \frac{\partial A}{\partial r} \frac{\partial B}{\partial r} + \frac{\partial^{2} B}{\partial r^{2}} + \frac{1}{r} \frac{\partial B}{\partial r}) = 0

We recall that there exists an exact analytical solution of Eq. (6), expressed in term of Kummer function [30, 31],

M_{n m}^{s o l} (Z, r, φ) = \frac{η}{ξ_{}} [\cos (m φ) + i q \sin (m φ)] {(\frac{r}{ξ_{}})}^{m} V_{n}^{(m)} (\frac{r^{2}}{ξ_{}^{2}}) e^{\frac{- r^{2}}{2 ξ_{}^{2}} + i b (z)}

where

{\begin{cases} η = \sqrt{n! / Γ (n + m + 1)} \\ V_{n}^{(m)} (θ) = [Γ (m + 1 + n) / (n! Γ (m + 1 + n))] F (- n, m + 1, θ) \end{cases}

here

Γ

and

F

are the Gamma function and the Kummer function, respectively. The parameter

q \in [0, 1]

determines the modulation depth of the beam intensity [32]. The beam width is represented by

ξ = ξ_{0} \sqrt{1 + Z^{2} / Z_{R}^{2}}

is the beam width at the propagation distance

Z

. Here,

ξ_{0}

is the beam waist of the fundamental or Gaussian mode

(n = 0, m = 0)

, and the Rayleigh length

Z_{R}^{}

is the unit of length in the propagation direction. And the phase offset of the beam

b (Z) = b_{0} - (2 n + m + 1) Z / ξ_{}^{2}

.

Figure 2 shows the transverse intensity distributions of $M_{n m}^{}$ with various parameters: radial mode number $n$ , azimuthal mode number $m$ and the modulation depth $q$ . One can clearly see that $M_{n m}^{}$ modes have some common characteristics determined by $n$ , $m$ and $q$ . Physically, the parameter $n$ is the number of radial nodes in the intensity distribution, while $m$ and $1 - q$ account for azimuthal angle distribution and azimuthal modulation depth, respectively. Taking Fig. 2(a2) as an example, with $n = 2$ , the corresponding number of concentric rings will form in the spatial profile, and the distribution of the intensity is obviously irrelevant for the azimuthal angle distribution for the reason $m = 0$ . In this case, when $m$ is increased to 1 and 2, the number of azimuthal index correspond to 1 and 2 as shown in Fig. 2(b2) and 2(c2), respectively, where the concentric rings are changed into azimuthally-modulated rings and $2 m$ determine the number of beads. Now we set $q = 1$ and keep $n = 1 (m = 2)$ , and redo the simulation presented in Fig. 2(e); in comparison with the result shown in Fig. 2(a2), we find that the beam profile also features ring-like structure, while appearing one singularity around the origin. Figure 2(a3)-2(c3) display the intensity distribution of the higher order modes $n = 3$ with different $m$ , while the parameter $q$ is set to 1 in Fig. 2 (f). They also comply the rule. The major difference from $q = 0$ and $q = 1$ is readily apparent; with $q = 0$ the topological charges of the solution $M_{n m}^{}$ disappears, while it at the origin still remains with various $n$ in the $q = 1$ case.

Fig. 2 Transverse intensity distribution of $M_{n m}^{}$ for different values of radial mode number $n$ , azimuthal mode number $m$ : (a1)-(c1) $n = 1$ , (a2)-(c2) $n = 2$ and (a3)-(c3) $n = 3$ corresponding to $m = 0$ , 1 and 2 with $q = 0$ , respectively. (d)-(f) $m = 2$ corresponding to $n = 0$ , 1 and 2 with $q = 1$ , respectively. The insets present phase profile.

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Combining Eqs. (5a) and (7), we finally obtain the exact analytical solution of Eq. (3)

\begin{matrix} ψ_{n m} (Z, R, φ, T) = \frac{η}{ξ_{}} A i (T - \frac{Z^{2}}{4} + i a Z) e^{a T - \frac{1}{4} a Z^{2} + i (Z - \frac{1}{24} Z^{3} + \frac{1}{2} a^{2} Z + \frac{1}{2} T Z)} \\ \times [\cos (m φ) + i q \sin (m φ)] {(\frac{r}{ξ_{}})}^{m} V_{n}^{(m)} (\frac{r^{2}}{ξ_{}^{2}}) e^{\frac{- r^{2}}{2 ξ_{}^{2}} - i b (Z)} \end{matrix}

3. Analysis and discussion

Having found the analytical solution of Eq. (9), we concentrate on the comparison of analytical solutions with numerical simulations of the (3 + 1)D NNLSE in the following. We discuss various examples of the localized wave packets for different parameters. The evolution properties of the Airy–Laguerre–Gaussian light bullets can be easily manipulated and modulated through adjusting the mode number $n$ and $m$ , and the modulation depth $q$ .

3.1. Zero–vorticity Airy–Laguerre–Gaussian light bullets ( $m = 0$ )

To guarantee high numerical precision, we utilize a fourth-order split-step fast Fourier transform (FFT) method [33] in double precision. 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 Fig. 3, we show the iso-surface plots of analytical solution Eq. (9) with $m = 0$ at propagation distance $Z = 0$ (the first column) and $Z = 5 Z_{R}$ (the second column) with different $n$ . As mentioned above, we predict that the parameter $n$ will account for the number of concentric rings in the intensity distribution. Specifically, with $n = 1$ ( $n = 2$ ), Fig. 3(a1) (Fig. 3(a2)) presents the initial beam profile of the light bullet at $Z = 0$ . It features that seven ellipsoidal pulses are stacked in the vertical direction, and the corresponding one (two) coaxial ring appears around the first, second, and third pulse in the horizontal ( $x, y$ ) plane. The reason is quite clear—it is determined by the radial mode number $n = 1$ ( $n = 2$ ). Along the accelerating direction $T$ , the Airy function distribution is for the reason that the intensity becomes weaker. However, with the propagation distance increasing to $Z = 5 Z_{R}$ , by comparing Fig. 3(b1) (Fig. 3(b2)) with Fig. 3(a1) (Fig. 3(a2)), owing to the modulation of the SNNM, it is clear that bullet becomes apparent compressed and appears more rings encircling the $T$ -axis. This phenomenon can be explained by the phase transition that an Airy profile is transformed into a Gaussian one at a critical point upon propagation due to the strong nonlocal nonlinear potential [34]. So, the bullet profile changes markedly during propagation. This propagation property of the bullet exhibited in SNNM is quite different from that in free space [17–20], in which such wave packet can retain its structure over several Rayleigh lengths during propagation. Note that the accelerating characteristic can also be verified by the pulse interval, ranging from $- 5$ to 5 at $Z = 5 Z_{R}$ , while at $Z = 0$ it ranges from $- 10$ to 0. At $Z = 5 Z_{R}$ , the numerical simulations correspond to $n = 1$ and $n = 2$ are displayed in Fig. 3(c1) and Fig. 3(c2), respectively. We find that the simulation results are in good agreement with our theoretical calculations.

Fig. 3 Snapshots describing the evolution of Airy–Laguerre–Gaussian light bullets with zero-vorticity ( $m = 0$ ) at $Z = 0$ (the first column) and $Z = 5 Z_{R}$ (the second and third column) with $n = 1$ (top row) and $n = 2$ (bottom row). Note that the second and third columns are obtained by analytically and numerically, respectively. Other parameters as chosen as follows: $a = 0.1$ , $γ = \sqrt{2}$ and $P_{0} = 1$ .

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3.2. Necklace Airy–Laguerre–Gaussian light bullets ( $q = 0, m \neq 0$ )

In the following we discuss the properties of necklace Airy–Laguerre–Gaussian light bullets. Figure 4 shows the spatiotemporal profile of the bullet with the mode number $n = m$ at different propagation distance. The major different from Fig. 3 is readily apparent; owing to the azimuthal modulation ( $m \neq 0$ ), the coaxial rings in each layer are transformed into necklace form with $2 m$ beads, and the intensity structure feature 3D necklace beams. Figure 4(a1) is the initial beam profile of the analytical solution at $Z = 0$ . With $n = m = 1$ , the corresponding intensity structure is fragmented: five layers in the vertical direction, and two ( $2 m$ ) necklace ellipsoids in the horizontal plane in each layer. Figure 4(b1) displays the pattern of the analytical solution of Eq. (9) at $Z = 5 Z_{R}$ , while Fig. 4(c1) illustrates the numerical simulation. With propagation, the modulation of the SNNM can lead the Airy pulse in time to change drastically and exist phase transition [34], which result in curious shape change that the light bullet becomes distinct compressed and appears more layers in the vertical direction. This behaviour is clearly different from the results of Refs [17] and [20], in which the bullet does not undergo a profound change in free space. In addition, the Fig. 4(a2)-4(c2), show another iso-surface contour plot of this localized necklace wave at different propagation distance, with $n = m = 2$ .

Fig. 4 Necklace Airy–Laguerre–Gaussian light bullets. Setup is the same as in Fig. 3, except for $m = 1$ (top row) and $m = 2$ (bottom row).

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3.3. Vortex Airy–Laguerre–Gaussian light bullets ( $q = 1, m \neq 0$ )

By setting parameters to be $q = 1$ and $m = 1$ in Eq. (9), we can get vortex Airy–Laguerre–Gaussian light bullets. Figure 5(a1) displays the initial beam profile of the analytical solution of Eq. (9) at $Z = 0$ . In comparison with those shown in Fig. 3 and Fig. 4, we find that the central ellipsoidal pulses change into rings. Because of the condition $q = m = 1$ , it results in the term $\cos (m φ) + i q \sin (m φ) = e^{i φ}$ in Eq. (9). Therefore, as depicted in Fig. 5(a2), the phase distribution appears one singularity (placed in white ellipse) that makes energy flow around it, while the intensity is equal to zero at singularity point, rotating counterclockwise. This is for the reason that along $T$ axis the intensity structure is characteristic of vortex. Figure 5(b1) and 5(c1) is the pattern of the analytical solution and the numerical simulation at $Z = 5 Z_{R}$ , which correspond to phase distribution shown in Fig. 5(b2) and Fig. 5(c2), respectively. The phenomenon can also be explained by the reason discussed above.

Fig. 5 (a1)-(c1) Vortex Airy–Laguerre–Gaussian light bullets with $n = 1$ , $m = 1$ and $q = 1$ at $Z = 0$ (the first column) and $Z = 5 Z_{R}$ (the second and third column). (b1) and (c1) are obtained by analytically and numerically, respectively. (a2)-(c2) The transverse phase distribution correspond to (a1)-(c1). The singularity is placed in white ellipse.

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4. Conclusion

In summary, we have introduced a model based on the spatiotemporal (3 + 1)D nonlinear Schrödinger equation, to consider propagation of spatiotemporal optical wave packets in SNNM, both analytically and numerically. As an eigenmode of 3D nonlinear Schrödinger equation, an exact solution in form of self-accelerating Airy–Laguerre–Gaussian nonlinear light bullets has been reported. By selecting different values of radial mode number, the azimuthal mode number and the modulation depth, the controllable behavior of 3D spatiotemporal wave packets are displayed, which may appear in the form of the disk-shaped fundamental pulses, the necklace rings and the vortex rings. In stark contrast to the linear one in free space, we find that the spatiotemporal profiles can be profoundly affected by nonlocality during propagation. Our results not only deepen the understanding of self-accelerating beams, but also widen their potential applications, such as optical communications, optical tweezing, the generation of plasma channels, and microlithography.

Funding

Key University Science Research Project of Henan Province (No. 17A140003; No. 2016YBZR042); Postdoctoral Research Sponsorship (No. 190308); Natural Science Foundation of Shaanxi Province (No. 2016JM1006); National Natural Science Foundation of China (Grant Nos. 11674002 and 11205001);. Postdoctoral Research Sponsorship in Henan Province (No. 2014035); Postdoctoral Research Sponsorship (No. BH2014033); Science and Technology Department of Henan Province (No. 144300510018).

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Self-accelerating Airy-Laguerre-Gaussian light bullets in a two-dimensional strongly nonlocal nonlinear medium

Abstract

1. Introduction

2. Theoretical model

3. Analysis and discussion

3.1. Zero–vorticity Airy–Laguerre–Gaussian light bullets ( $m = 0$ )

3.2. Necklace Airy–Laguerre–Gaussian light bullets ( $q = 0, m \neq 0$ )

3.3. Vortex Airy–Laguerre–Gaussian light bullets ( $q = 1, m \neq 0$ )

4. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (12)

Optics Express

Abstract

1. Introduction

2. Theoretical model

3. Analysis and discussion

3.1. Zero–vorticity Airy–Laguerre–Gaussian light bullets (m=0)

3.2. Necklace Airy–Laguerre–Gaussian light bullets (q=0,m≠0)

3.3. Vortex Airy–Laguerre–Gaussian light bullets (q=1,m≠0)

4. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (12)

Optics Express

3.1. Zero–vorticity Airy–Laguerre–Gaussian light bullets ( $m = 0$ )

3.2. Necklace Airy–Laguerre–Gaussian light bullets ( $q = 0, m \neq 0$ )

3.3. Vortex Airy–Laguerre–Gaussian light bullets ( $q = 1, m \neq 0$ )