Propagation dynamics of self-accelerating second-order Hermite complex-variable-function Gaussian wave packets in a harmonic potential

Jingyun Ouyang; Dongmei Deng; Dongmei Deng; Xi Peng

doi:10.1364/OE.517127

1. Introduction

Berry and Balazs initially proposed the Airy solution [1] to the Schrödinger equation without an outfield in 1979. Airy beam is a specific type of optical beam that possesses unique capabilities [2] of non-diffracting, self-reconstruction, and self-accelerating. These advantageous propagation characteristics have paved the way for their applications in various scenarios [3]. Research in this field has extended from the investigation of one-dimensional pulses [4] and two-dimensional beams [5,6] to the exploration of three-dimensional wave packets [7–10]. Specifically, recent studies have focused on unraveling the propagating morphology of self-accelerating wave packets [11] and understanding the reversal of their self-accelerating direction.

The cross-phase has discovered numerous applications in modern optics and other fields. Changing the cross-phase factor of Hermite-Gaussian modes allows them to transform into Laguerre-Gaussian modes, and vice versa [12]. In the field of autofocusing, low-order cross-phase is used to shape the intensity distribution of circular Airy vortex, demonstrating its asymmetric modulation capability [13]. Furthermore, the cross-phase factor can flexibly adjust the focal length of the ring Pearcey beam, its focusing ability, and the beam focus direction [14]. Additionally, the cross-phase enables stable propagation of linear and nonlinear evolution of the spiraling elliptic hollow beams [15]. The effects of cross-phase on focal depth and simultaneous capture of multiple particles have also been studied. In the field of wave packets, the cross-phase factor can achieve more precise modulation over wave packets. Shi et al. designed inseparable phases to enhance control accuracy and freedom of optical manipulation [16].

Standard Hermite-Gaussian beam [17] is a classic beam model used by optical researchers to describe high-order Gaussian beams. It is intriguing to explore the flexible adjustment of spatiotemporal wave packets through the use of cross-phase. In this article, we focus on introducing cross-phase to self-accelerating second-order Hermite complex-variable-function Gaussian (SSHCG) wave packets to investigate wave packet shaping and reverse acceleration during propagation. The SSHCG wave packets propagating in a harmonic potential is researched, and we emphasise on studying the influence of the distribution factor, the cross-phase, the potential depth, and the chirp parameter on the SSHCG wave packets.

The structure of the paper is as follows. In Sec. 2, the theoretical model of the (3+1)-dimensional SSHCG wave packets under the harmonic potential is studied. Then in Sec. 3, the propagation properties of spatiotemporal SSHCG wave packets are discussed. In addition, the energy flow and the angular momentum of the wave packets are investigated in Sec. 4. What’s more, the radiation force on a nanoparticle is reported in Sec. 5. Last but not least, we summarise the paper in Sec. 6.

2. Theoretical model

In a harmonic potential [18], the propagating evolution of the optical spatiotemporal wave packets is described by the Schrödinger equation [19–21]

(1)$$2i\frac{\partial \psi}{\partial z}+\frac{1}{k}(\frac{\partial^{2}\psi}{\partial x^{2}}+\frac{\partial^{2}\psi}{\partial y^{2}})-k_{g}\frac{\partial^2\psi}{\partial t^2}-k\beta^{2}(x^{2}+y^{2})\psi=0,$$

where $\psi (x,y,t,z)$ denotes the complex envelop of the optical field, the wave number $k(\omega _{0})=n\omega _{0}/c$ and the group velocity dispersion $k_{g}(\omega _{0})=\partial ^{2}k/\partial \omega _{0}^{2}$ are evaluated at the carrier frequency $\omega _{0}$. $\beta$ is a factor that regulates the depth of potential. $w_{0}$ is the spatial scaling factor, $t_{0}$ is the temporal scaling factor, $z_{R}=kw_{0}^{2}$ is the diffraction length. Here, the solution of $\psi (x,y,t,z)$ can be written in following multiplication

(2)$$\psi(x,y,t,z)=U(x,y,z)A(t,z).$$

With the dispersion length $L_{disp}=t_{0}^{2}/\mid k_{g} \mid$ being considered, we assume that the dispersion and diffraction have the same quantitative effect along the propagation (i.e. $z_{R}=L_{disp}$) [21]. After substituting Eq. (2) into Eq. (1), and using the separation of variables method [8,9], we obtain the following two equations

(3)$$2i\frac{\partial A(t,z)}{\partial z}-k_{g}\frac{\partial^2 A(t,z)}{\partial t^2}=0,$$

(4)$$2ik\frac{\partial U(x,y,z)}{\partial z}+\frac{\partial^2 U(x,y,z)}{\partial x^2}+\frac{\partial^2 U(x,y,z)}{\partial y^2} -k^{2}\beta^{2}(x^{2}+y^{2})U(x,y,z)=0.$$

We study the self-accelerating chirped Airy function case in Eq. (3) where $A(t,0)=Ai(t/t_{0})\exp [at/t_{0}-ic_1 t/t_{0}-ic_2 t^2/t_{0}^{2}]$ [22,23]. Here, $Ai(t/t_{0})$ represents the Airy function, where its primary lobe is situated at the leading edge. Parameter $a$ ($0<a\le 1$) introduces an attenuation factor along the $t$ direction, making it possible for the physical realization of finite energy. Additionally, $c_{1}$ and $c_{2}$ are chirp modulation parameters corresponding to the first and second orders, they have the capability to control the direction of the primary lobe. Under these initial conditions, Eq. (3) can be solved in the following manner

(5)$$\begin{aligned}A&\left (t,z \right )=\sqrt{\zeta }Ai\left [ \zeta \left ( \frac{t}{t_{0}}-c_{1}\frac{z}{z_{R}}-\frac{\zeta}{4}(\frac{z}{z_{R}})^{2}-ia\frac{z}{z_{R}} \right ) \right ]\\ &\times \exp\left [ a\zeta \left (\frac{t}{t_{0}}-c_{1}\frac{z}{z_{R}}-\frac{\zeta}{2}(\frac{z}{z_{R}})^{2} \right ) \right ]\\ &\times \exp\left [{-}i\zeta\left ( c_{1}\frac{t}{t_{0}}+c_{2}(\frac{t}{t_{0}})^{2}+\frac{\zeta tz}{2t_{0}z_{R}}+\frac{a^{2}z}{2z_{R}}-\frac{c_{1} ^{2}z}{2z_{R}} -\frac{c_{1}\zeta}{2} (\frac{z}{z_{R}})^{2}-\frac{\zeta^{2} }{12}(\frac{z}{z_{R}})^{3} \right ) \right ], \end{aligned}$$

where $\zeta =1/\left ( 1+2c_{2}z/z_{R} \right )$, this parameter determines the adjusted positioning of the envelope of the Airy function. The Airy part can reverse when $\zeta <0$, placing its tail (side lobes) in the front.

For Eq. (4), the initial second-order Hermite complex-variable-function Gaussian distribution with cross-phase can be expressed as

(6)$$\begin{aligned}U(x,y,0)&=H_{j}\left ( \frac{x_{0}+iy_{0} }{bw_{0}} \right )\exp\left ( -\frac{ x_{0}^{2}+y_{0}^{2} }{2w_{0}^{2}} \right )\\ &\times \exp\left [ \frac{iu}{w_{0}^{2}}\left ( x_{0}\cos \theta -y_{0}\sin \theta \right ) \left (x_{0}\sin \theta +y_{0}\cos \theta \right ) \right ], \end{aligned}$$

where $H_{j}( \cdot )$ is a Hermite complex-variable-function [24], and the analysis here is conducted for the case of $j=2$, $b$ relates to the distribution coefficient, $u$ is the real coefficient that influences how strongly the cross-phase modulation effect occurs [12], $\theta$ characterizes the rotation angle within a specific plane. At $\theta =0$, the low-order cross-phase can be simplified to $\exp ( iux_{0}y_{0}/w_{0}^{2})$, and we only consider this typical case.

After performing some calculations, we arrive at the solution for Eq. (4) as follows

(7)$$U(x,y,z)=\left ( \frac{x^{2} }{w_{1}^{2} }-\frac{y^{2} }{w_{2}^{2} }+iMxy-N \right )\exp\left ( -\frac{x^{2}+y^{2} }{w_{3}^{2} }+i\Theta xy \right ),$$

where $w_{1}^{2}=iBb^{2}w_{0}^{2}\sqrt {A_{1}A_{2}}(4A_{1}^{2} +u ^{2}) ^{2}/[2( 2A_{1}-u)^{2}]$, $w_{2}^{2} =iBb^{2}w_{0}^{2}\sqrt {A_{1}A_{2} }( 4A_{1} ^{2}+u^{2} )^{2} /[2( 2A_{1}+u)^{2}]$, $w_{3}^{2} =(1+2iA_{2}BD)w_{0}^{2}/(4A_{2}B^{2})$, $A_{1}=(B+iA)/(2B)$, $A_{2}=A_{1}+u^{2}/(4A_{1})$, $M=(u^{2}-2A_{1}A_{2})/(2A_{1}^{2}A_{2}^{2}B^{2}w_{0}^{2})$, $\Theta =-u/(4A_{1}A_{2}B^{2}w_{0}^{2})$, $N=-i(u+A_{1}A_{2}b^2)/(\sqrt {A_{1}^3 A_{2}^3}Bb^2)$, $A=\cos (z/\beta )$, $B=\beta \sin (z/\beta )$, and $D=-\sin (z/\beta )/\beta$.

The solutions of spatiotemporal SSHCG wave packets corresponding to Eq. (1), can be constructed in the following manner using the expressions provided in Eqs. (5) and (7) as

(8)$$\begin{aligned}\psi&\left (x,y,t,z\right )=\left ( \frac{x^{2} }{w_{1}^{2} }-\frac{y^{2} }{w_{2}^{2} }+iMxy-N \right ) \exp\left ( -\frac{x^{2}+y^{2} }{w_{3}^{2} }+i\Theta xy \right )\\ &\times\sqrt{\zeta }Ai\left [ \zeta \left ( \frac{t}{t_{0}}-c_{1}\frac{z}{z_{R}}-\frac{\zeta}{4}(\frac{z}{z_{R}})^{2}-ia\frac{z}{z_{R}} \right ) \right ]\\ &\times \exp\left [ a\zeta \left (\frac{t}{t_{0}}-c_{1}\frac{z}{z_{R}}-\frac{\zeta}{2}(\frac{z}{z_{R}})^{2} \right ) \right ]\\ &\times \exp\left [{-}i\zeta\left ( c_{1}\frac{t}{t_{0}}+c_{2}(\frac{t}{t_{0}})^{2}+\frac{\zeta tz}{2t_{0}z_{R}}+\frac{a^{2}z}{2z_{R}}-\frac{c_{1} ^{2}z}{2z_{R}} -\frac{c_{1}\zeta}{2} (\frac{z}{z_{R}})^{2}-\frac{\zeta^{2} }{12}(\frac{z}{z_{R}})^{3} \right ) \right ]. \end{aligned}$$

3. Spatiotemporal SSHCG wave packets

To gain a deeper understanding of SSHCG wave packets, we begin by examining their characteristics within a harmonic potential.

Figure 1 depicts the propagation trajectory of the SSHCG wave packet, which remains unaffected by cross-phase ($u=0$). It also shows the butterfly shapes of spatiotemporal localized SSHCG wave packets at different propagation distances with $b=2$. The SSHCG wave packet exhibits a twisted rotational state with good central symmetry in the intensity distribution, and it rotates uniformly clockwise with increasing distance. Figures 1(b)-(e) display the spatiotemporal localized SSHCG wave packet corresponding to the blue line segment in Fig. 1(a). The size of the SSHCG wave packet remains constant, while the inserted part in the bottom right corner shows two vortexes phase distribution of that component. As the propagation distance increases, the spatiotemporal localized SSHCG wave packet accelerates in the $t$ direction.

Fig. 1. (a) Propagation trajectory of SSHCG wave packets. (b)-(e) SSHCG wave packets with different propagation distances. $b=2$, $u=0$, $\beta =1$, $c_{1}=1.5$, and $c_{2}=0$.

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Figure 2 presents the analytical solution of Eq. (8), demonstrating the propagation trajectory of the SSHCG wave packets under the influence of the cross-phase. The changing images of spatiotemporal localized SSHCG wave packets at different propagation distances are also shown. Compared to Fig.1, the SSHCG wave packets display varying effects on beam shaping at different propagation distances within the same period. Unlike the gradually diverging energy characteristics of a uniform medium, the range of periods in a harmonic potential is defined by $\beta$. In Figs. 2(b)-(e), one observes that the wave packet corresponding to different propagation distances shows positive acceleration along the $t$ direction. Within a single period, the wave packet undergoes expansion in the first half-period and contraction in the other half-period in addition to its uniform rotation, leading to the self-stretching phenomenon. Utilizing phase information can reveal the characteristics brought about by the cross-phase more clearly. In Fig. 1(d) and Fig. 2(d), at $z/z_{R}=\pi /2$, the rotation angle changes to $90$ degrees compared to the initial position of the wave packet.

Fig. 2. (a) Propagation trajectory of SSHCG wave packets, (b)-(e) SSHCG wave packets with different propagation distances. $b=2$, $u=1$, $\beta =1$, $c_{1}=1.5$, and $c_{2}=0$.

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Figure 3 illustrates the spatiotemporal SSHCG wave packets, accompanied by the transverse intensity and phase distributions at $z/z_{R}=\pi /2$. The SSHCG waveform exhibits distinct configurations as the value of $b$ changes, such as: three peaks shape, two polarity shape, and elliptical shape. These configurations include a waveform divided into three parts by double dark cores (as illustrated in Fig. 3(a1)), a waveform splits into two parts by a dual dark core (as depicted in Fig. 3(b1)), and an overall waveform affected by the dual dark core but not split (as shown in Fig. 3(c1)). In Fig. 3(a2), the intensity distribution forms a structure consisting of three prominent peaks, characterized by two isolated dark spots, each carrying one topological charge. As $b$ increases, the dual dark cores of the SSHCG wave packet disperse towards both ends, while the central peak of the wave packet gradually concentrates and expands in Fig. 3(b2) and Fig. 3(c2). Upon reaching a specific threshold, the dual dark cores move away from each other toward the left and right, causing the waveform gradually converge around the center until it approaches the limit of a transverse Gaussian distribution $\left ( b\longrightarrow \infty \right )$. At this point, the waveform is no longer influenced by the dark cores. Different phase distributions with various $b$ are shown in Figs. 3(a3)-(c3). The distribution factor $b$ plays a crucial role in the formation process of the SSHCG wave packet.

Fig. 3. (a1)-(c1) Snapshots describing the SSHCG wave packets with different $b$. (a2)-(c2) The normalized intensity. (a3)-(c3) The phase distribution. $z/z_{R}=\pi /2$, $u=1$, $\beta =1$, $c_{1}=1.5$, and $c_{2}=0$.

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Figure 4 depicts SSHCG wave packets under various cross-phase factors and chirp parameters, showing their diverse range of shapes including three peaks shape and ring-shaped double-vortex structures. The cross-phase factor affects the phase distribution and influences the shape of the wave packet. Figures 4(a1)-(c1) and Figs. 4(a2)-(c2) maintain their main lobe at the front position. When altering the sign of the cross-phase factor under the same set of parameters, Figs. 4(a1)-(c1) and Figs. 4(a2)-(c2) show the variation in the SSHCG wave packet while maintaining a similar shape, but with opposite rotation directions. Consequently, it permits more precise control and shaping of SSHCG wave packets. In Figs. 4(a3)-(c3), we observe that swapping the values of the chirp parameters $c_{2}$ to negative results in the SSHCG wave packet exhibiting opposite characteristics, with sidelobes situated in the forward direction.

Fig. 4. The SSHCG wave packets under different cross-phase factors and chirp parameters. $c_{1}=1.5$ and $c_{2}=0$ in the first and second rows, $c_{1}=0$ and $c_{2}=-1.5$ in the third row. $b=0.8$, $z/z_{R}=\pi /3$, and $\beta =1$.

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4. Energy flow and angular momentum

Furthermore, a discussion on the rotation of the wave packet is essential to further understand the propagation dynamics of SSHCG. The Poynting vector is a crucial basis for studying the energy transfer of optical beams, providing profound insights into the characteristics of energy flow. It represents the energy flow density vector within the electromagnetic field, signifying the magnitude represents the power per unit area carried by the electromagnetic wave, and its direction indicates the direction of energy flow. Given a vector potential as $\vec {\varepsilon }U\left (x,y,z \right )e^{-ikz}$, where $\vec {\varepsilon }$ represents an arbitrary polarization. In the Lorentz gauge, if we consider a polarization field oriented along the unit vector $\vec {x}$ direction, the expression for the time-averaged Poynting vector [19,24,25] can be formulated as follows

(9)$$\vec{\left \langle S \right \rangle}=\frac{c}{4\pi }\left \langle \vec{E}\times \vec{B} \right \rangle=\frac{c}{8\pi}\left [ i\omega \left (U\nabla _{\bot}U^{{\ast} }-U^{{\ast} }\nabla _{\bot}U \right )+2\omega k\left | U \right |^{2}\vec{e}_{z} \right ],$$

where $\nabla _{\bot }=\frac {\partial }{\partial _{x} }\vec {e}_{x}+\frac {\partial }{\partial _{y} }\vec {e}_{y}$, the unit vectors $\vec {e}_{y}$ and $\vec {e}_{z}$ point in the $y$ and $z$ directions, respectively.

In Fig. 5, we show SSHCG wave packet with two vortexes, the simulation of energy flow in a harmonic potential. The background highlights the overall energy distribution while the red arrows represent the transverse energy flow. The magnitude and direction of the red arrows correspond to the magnitude and direction of the transverse plane energy flow. It is noteworthy that the area covered by the red arrows, consistent with the background, indicates that transverse energy flow plays a crucial role in the overall energy distribution. Notably, the rotation direction of the transverse energy flow changes due to the positive or negative cross-phase factor.

Fig. 5. Transverse energy flow (red arrows) and normalized energy flow (background) of SSHCG wave packets. First row with $b=0.8$, $\beta =1$, and $u=-3$. Second row with $b=1.3$, $\beta =0.5$, and $u=3$.

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The time averaged angular momentum [24,25] can be expressed in physics as

(10)$$\begin{aligned}\vec{\left \langle J \right \rangle } &=\vec{r}\times \left \langle \vec{E} \times \vec{B} \right \rangle\\ &=\frac{\omega}{2}\left [ \left ( 2yk\left | U \right | ^{2} -izS_{y} \right )\vec{e}_{x}+\left ( izS_{x} -2xk\left | U \right |^{2} \right ) \vec{e}_{y}+i\left ( xS_{y}-yS_{x} \right ) \vec{e} _{z} \right ] , \end{aligned}$$

in this formula, $S_{x} =U\frac {\partial U ^{\ast } }{\partial x} -U^{\ast } \frac {\partial U}{\partial x}$ and $S_{y}=U\frac {\partial U^{\ast } }{\partial y}-U^{\ast } \frac {\partial U}{\partial y}$.

The alteration in angular momentum corresponds to the torque induced by a variation in linear momentum. In Fig. 6, the red arrows indicate the direction of angular momentum density within the transverse plane, while the background scale represents the distribution density of total angular momentum. As shown in Figs. 5(a1)-(a4) and Figs. 6(a1)-(a4), the SSHCG wave packet exhibits a $180$-degree rotation characteristic in both energy flow and angular momentum as the propagation distance ranges from $0$ to $\pi$. When the potential depth $\beta$ is reduced by half, as seen in Figs. 5(b1)-(b4) and Figs. 6(b1)-(b4), the propagation distance is halved, making it similar to the initial propagation situation. In a harmonic potential, the SSHCG wave packet exhibits consistent rotational characteristics in energy flow and angular momentum.

Fig. 6. Transverse angular momentum density flow (red arrows) and normalized angular momentum density (background) of SSHCG wave packets. First row with $b=0.8$, $\beta =1$, and $u=-3$. Second row with $b=1.3$, $\beta =0.5$, and $u=3$.

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5. Radiation force

This section explores the gradient force and the scattering force during the propagation of SSHCG wave packets. By manipulating the parameters to modify the spatial configuration of the SSHCG wave packets, we can achieve functionalities such as beam expansion and rotation, enabling precise manipulation and control of particles [26–28].

If we consider a stable state of non-absorbing Rayleigh dielectric particle characterized by a refractive index of $n_{1}$, the expressions of the scattering force and the gradient force [26–29] are given by

(11)$$\vec{F}_{scat} =\frac{8\pi n_{2} }{3c} k^{4}\delta^{6} \left ( \frac{m^{2}-1 }{m^{2}+2} \right )^{2}I\vec{e}_{z},$$

(12)$$\vec{F}_{grad}=\frac{2\pi n_{2}\delta ^{3} }{c}\left ( \frac{m^{2}-1 }{m^{2}+2} \right ) \nabla I,$$

where $m=n_{1}/n_{2}$ is the relative refractive index, $n_{2}$ is the refractive index of the surrounding medium, $\delta$ represents the radius of the particle. The formula reveals that the magnitude of the radiation forces depends on various factors, including particle size, refractive index of the particle, refractive index of the surrounding medium, and light intensity distribution.

We set the surrounding medium as water $n_{2}=1.332$, assume that $n_{1}=1.592$, $\delta =15nm$, and $c=3\times 10^{8}m/s$.

The discussion here encompasses both the gradient force and the scattering force. When a particle is placed in a gradient field, this leads to greater pressure in regions with higher light intensity, causing the object to move towards the direction of higher light intensity to achieve optical capture [28]. Figure 7 illustrates the behavior of the scattering forces and the gradient forces during the propagation over a certain distance. By comparing Figs. 7(a2)-(d2) with Figs. 7(a1)-(d1), one can observe that the uneven distribution of intensity leads to uneven radiation force distribution exerted by the wave packet on the particle.

Fig. 7. (a1)-(d1) The scattering force. (a2)-(d2) The gradient force. $b=0.5$, $u=2$, and $\beta =1$.

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The prerequisite for particle capture is that the gradient force is greater than the scattering force, with $R=\frac {\left | F_{grad} \right | }{\left | F_{scat} \right | }\ge 1$ being the stability criterion. In Fig. 8, both the increase and decrease of the scattering force and the gradient force remain consistent in the same propagation distance, and the gradient force consistently remains an order of magnitude larger than the scattering force. Comparing Figs. 8(a) and 8(b), it is evident that changing the propagation distance does not affect the particle trapping ability. In Figs. 9(a) and 9(b), as the distribution factor increases, the scattering force gradually decreases, and the gradient force increases, leading to a corresponding increase in the $R$ value and enhancing the stability of light capture. Although both the scattering force and the gradient force of SSHCG wave packets change with the variation of the distribution factor, it always reaches the stability criterion. Overall, using SSHCG wave packets for particle capture holds good potential.

Fig. 8. The scattering force (a) and the gradient force (b) at different propagation distances. $b=0.5$, $\beta = 1$, and $u=2$.

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Fig. 9. The scattering force (a) and the gradient force (b) under different distribution factors. $z/z_{R}=\pi /3$, $\beta =1$, and $u=2$.

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6. Conclusion

In summary, this study investigates the SSHCG wave packets through the resolution of the spatiotemporal Schrödinger equation in a harmonic potential. The manipulation of SSHCG wave packets is demonstrated by adjusting the distribution factor, the cross-phase factor, and the chirp parameter during propagation. As the distribution factor increases, the SSHCG wave packets become more tightly concentrated around its central region. The cross-phase factor modifies the shape of the wave packet during propagation, resulting in expansion and contraction, leading to the self-stretching phenomenon. The second-order chirp parameter can alter the direction of the main lobe. The potential depth can regulate the rotational periodicity of the SSHCG wave packets. Furthermore, the rotational Poynting vector and the angular momentum are also investigated. Additionally, the impact of the propagation distance and distribution factors on optical gradient forces is explored. By comparing the magnitudes of the radiation force, increasing the distribution factor enhances the stability of particle capture by SSHCG wave packets. The theoretical framework and numerical results outlined in this paper will enhance our comprehension of SSHCG wave packets and their potential applications, particularly in the optical trapping of non-absorbing nanoparticles.

Funding

National Natural Science Foundation of China (11947103, 12004081, 12174122); Basic and Applied Basic Research Project of Guangzhou City (2023A04J0041); Talent Introduction Project Foundation of Guangdong Polytechnic Normal University (2021SDKYA142).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Propagation dynamics of self-accelerating second-order Hermite complex-variable-function Gaussian wave packets in a harmonic potential

Abstract

1. Introduction

2. Theoretical model

3. Spatiotemporal SSHCG wave packets

4. Energy flow and angular momentum

5. Radiation force

6. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (12)

Optics Express