Azimuthally Spliced Power-Exponential Phase Modulation for Focal Spot Shaping of Circular Airy Beams

Abstract

Circular airy beam (CAB) is a kind of new structured light with non-diffracting, self-focusing, and self-healing properties. Due to its wide applications, recently, numerous researchers have used various methods to modulate this kind of beam. We theoretically verify and experimentally demonstrate the azimuthal modulation method to shapes the focal spot of the CAB by modulating the CAB with the azimuthally spliced power-exponential phase. The results show that after modulating by an azimuthally spliced power-exponential phase, multi-focal spots can be generated on the self-focusing focal plane of the modulated CAB, and the number of the focal spots can be precisely controlled by controlling the number of segments of the spliced power-exponential phase. The situations of generating three, four, and five focal spots can be achieved via appropriate azimuthally spliced power-exponential phase modulation. We also calculate the intensity distribution, energy flow density, angular momentum density, and optical force of the modulated beam after tight focusing. The results illustrate the theoretical possibility of stable multiparticle trapping by the modulated beam. Our results pave the way for on-demand shaping of the self-focusing focus of the CAB, which will facilitate related applications, such as CAB based multi-particle trapping.

1. Introduction

Recently, considerable interest has been exhibited in the investigation of airy beams [1]. Due to the properties of self-accelerating, curved parabolic geometry, and self-healing [2], the airy beams have shown unique advantages in many fields, such as femtosecond laser filamentation [3], particle manipulation [4], high-resolution micro-imaging [5], and laser machining [6]. Circular airy beam (CAB) is a special type of airy beam with circular symmetry and airy envelope distribution in the radial direction [7], which evolves radially along a curved orbit and can achieve autofocusing without relying on any nonlinear effects as well as lenses. It is well known that controlling the propagation of an intense light bullet in a transparent medium is not a simple task, due to the fact that, in the propagation process, some nonlinear effects, such as the Kerr effect, multi-photon absorption, and ionization, can cause the optical wave packet to produce complex and uncontrolled remodeling [8]. The unique autofocus effect of the CAB allows it to maintain a very low light intensity before focusing and then rapidly focus at the focal spot to significantly increase the light intensity, which can avoid unnecessary nonlinear effects during the propagation process before the focusing; thus, it is an ideal candidate for transmitting high-energy wave packets in transparent media. Due to this, the CAB has been used to produce intense light-bullet wave packets [8], which can provide significant advantages in many fields, such as higher-order harmonic generation, attosecond physics, and precise micromachining of materials. In addition, the auto-focusing characteristics of the CAB can also be used to realize some novel optical manipulation applications [9]. For example, using a CAB, particles may be trapped in the primary ring of the CAB and propelled towards the focus along a curved trajectory, and subsequently, the particles can be further accelerated along a straight path due to the weak diffraction characteristics of the light field at the focus [10]. Moreover, by properly adjusting the focal spot of the CAB, stable 3D trapping can be realized at the focal spot, and recent experimental comparison studies show that the CAB can almost keep the optical trapping stiffnesses constant at different optical trapping depths compared with the Gaussian beam under the condition of the same focal spot size. This means that the optical trapping generated by a CAB is robust against spherical aberration.

However, in previous works, the used CAB is usually unmodulated, which has only one single self-focusing spot. Further shaping of the self-focusing spot to achieve the precise control of structured distribution of multi-focal spots from the original single focal spot can provide more adjustable degrees of freedom for related applications and provide new opportunities for further technological development and breakthroughs in its application fields. For example, in the generation of strong light light-bullet wave packets, the unmodulated CAB can only produce a single wave packet, while the CAB with precise focal spot regulation can produce a customized wave packet array, and the shape of the wave packet can also be regulated. Another example is that in optical manipulation, unmodulated CAB can only trap single particles, whereas the modulated CAB enables multiparticle trapping and their array assembly. Therefore, it is of great practical significance to study the focus modulation method of CAB, which can provide technical support for diversified on-demand shaping of self-focusing spots in practical applications. Although the focal spot modulation of CAB is of great interest, to achieve flexible and diverse on-demand shaping of the self-focusing focal spot remains difficult and requires in-depth research [11].

Since CAB and airy beams are very unique structured light beams themselves, in the recent research frontier, a great deal of research works is still focused on their own generation, transmission, and applications. Examples include airy-beam-based tomography microscopy [12], THz-band airy beam generation for imaging via silicon diffracttion devices [13], airy beam generation via metasurxface [14], realization of 3D trapping via CAB [11], and circular Airy vortex beams via a uniaxial crystal separate radial-and azimuthal-polarization components [15]. In recent years, however, more and more researchers have paid attention to the development of more tunable degrees of freedom to realize multidimensional regulation of CAB. Zhao et al. applied a power-exponent phase to CAB to obtain a helically shaped autofocusing spot [16]; Wang et al. applied a cosine phase to CAB to produce multiple bladed spots in the angular direction [17]; Apostolos Brimis et al. [18] proposed a tornado-like spiral-focused beam called tornado waves by superimposing two CABs with different parabolic transmission paths and different helical phases, which was experimentally demonstrated subsequently [19,20]; Pan et al. constructed a galaxy wave satisfying the SU(2) coherent state by superimposing different CABs, which can achieve the evolutionary control of three-dimensional non-uniform rotational angular velocity of the self-focusing focus [21]; Liu et al. modulated CAB in Fourier space to obtain multiple focal spots [22]; and Lin et al. applied a segmented vortex phase to CAB to cause the beam to split and form multiple focal spots for image enhancement [23]. Although these studies provide profound insights for achieving diversified regulation of CAB, the light intensity distributions at the focal plane of these modulated CAB are very complex, and the shape of their focal spots is irregular. These complex and irregular focal spot intensity distributions are actually not conducive to practical applications. For example, in the application of intense light bullets wave packets generation, the irregular focal spot will lead to energy dispersion; in the application of laser processing, the irregular focal spot is not conducive to precise micromachining; and in the application of optical manipulation, the irregular focal spot is not conducive to the stable particles trapping and manipulation. In this regard, how to generate multiple independent, regular focal spots in the focal plane of a CAB by applying appropriate modulation is still an unsolved problem that deserves further study.

Therefore, in this work, we propose a simple method for shaping the focal spot of CABs. It is found that when a CAB carries an azimuthally spliced power-exponential phase, multiple independent and regular focal spots can be generated at the focal plane. Via appropriate azimuthally spliced power-exponential phase modulation, the generation of three, four, and five focal spots are demonstrated both theoretically and experimentally. Moreover, to promote its application in multi-particle trapping, the particle-trapping performance of the CAB carrying azimuthally spliced power-exponential phase is also discussed via calculating its gradient force, scattering force, intensity distribution, transverse power current density, and spin angular momentum after tight focusing. Our results provide an opportunity to flexibly regulate the focal spot of CAB and may have potential application value in the field of multi-particle trapping.

2. Propagation Characteristics of the Modulated CAB in Free Space

2.1. Theoretical Background

The electric field expression of the CAB modulated by azimuthal function in cylindrical coordinate system is as follows:

$$E\left(r,\phi ,0\right)=CAi({\displaystyle \frac{r_0-r}{w}})\exp (a{\displaystyle \frac{r_0-r}{w}})f(\phi ),$$

(1)

where Ai(·) denotes the airy function, with $r_0,w,a$ representing the radius of the airy beam, the scale factor, and attenuation coefficient, respectively. C is a constant representing the amplitude, and $(r,\phi ,0)$ is the cylindrical coordinate system with z = 0. The azimuthal function can be expanded into a superposition of a series of standard helical phases [24,25], i.e., $f(\phi )={\displaystyle \sum _{n=-\infty }^{\infty }{F}_n\exp (in\phi )},$ $F_n$ is the Fourier expansion coefficient, where $F_n={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_0^{2\pi }f(\phi )\exp (-in\phi )}d\phi$.

Thus, Equation (1) can then be rewritten as follows:

$$E\left(r,\phi ,0\right)={\displaystyle \sum _{n=-\infty }^{\infty }{F}_n}CAi({\displaystyle \frac{r_0-r}{w}})\exp (a{\displaystyle \frac{r_0-r}{w}})\exp (in\phi ).$$

(2)

During the propagation, the expression of the propagation field of the modulated CAB is given by the following equation [22,26]:

$$\begin{array}{l}E\left(r,\phi ,z\right)={\displaystyle \frac{1}{2\pi }}{\displaystyle \sum _{n=-\infty }^{\infty }\{F_n}\exp (in\phi ){\displaystyle {\int }_0^{\infty }{\stackrel{\sim }{Q}}_n\left(k_{\perp }\right)}\exp \left[iz\sqrt{k^2-k_{\perp }^2}\right]J_n\left(k_{\perp }r\right)k_{\perp }d{k}_{\perp }\}\\ {\stackrel{\sim }{Q}}_n\left(k_{\perp }\right)={\displaystyle \frac{C{w}^2}{2\pi }}({\displaystyle \frac{r_0}{w}}+w^2{k_{\perp }}^2)\sqrt{{\displaystyle \frac{\left(w^2{k_{\perp }}^2/3+k_{\perp }{r}_0\right)}{w^2{k_{\perp }}^2+k_{\perp }{r}_0}}}\exp \left(-a^{2}w{k}_{\perp }\right)J_n\left({\displaystyle \frac{w^2{k_{\perp }}^2}{3}}+k_{\perp }{r}_0\right),\end{array}$$

(3)

where k is the wave vector, and $k_{\perp }=\sqrt{k_x^2+k_y^2}$ is the transverse wave vector.

In this article, we use the azimuthally spliced power-exponential phase to modulate the CAB. Its function expression is described as follows:

$$f(\phi )=\left\{\begin{array}{l}\exp \{il\phi +i\alpha {[(\phi -\pi /3)/2\pi ]}^2\},0\le \phi <2\pi /3\\ \exp \{il\phi +i\alpha {[(\phi -\pi )/2\pi ]}^2\},2\pi /3\le \phi <4\pi /3\\ \exp \{il\phi +i\alpha {[(\phi -5\pi /3)/2\pi ]}^2\},4\pi /3\le \phi <2\pi .\end{array}\right.$$

(4)

2.2. Results and Discussions

The modulation function $f(\phi )$ consists of a standard helical phase and three segments of the spliced power-exponential phase. Considering l = 1 and $\alpha =-20\pi$, Figure 1(a1) shows the phase distribution of the spliced power-exponential phase, Figure 1(a2) is the phase distribution of the spiral phase, and Figure 1(a3) shows the modulation function $f(\phi )$ phase distribution. Figure 1(a4) shows the OAM spectrum, due to the fact that $f(\phi )$ can be expanded into a superposition of a series of standard helical phases, so the relative power of the nth-order OAM mode can be obtained by $P_n={\left|F_n\right|}^2$. It can be seen that the power is mainly concentrated in −10~10th order OAM modes (the power of OAM modes beyond this range is very low). In the calculation, we consider only the main angular momentum spectral components that concentrate most of the energy of the beam. The motivation of designing such a phase modulation function is clear: the azimuthally spliced power-exponential phase term forms three parabolic phase distributions in the three sectors of $0<\phi \le 2\pi /3, 2\pi /3<\phi \le 4\pi /3$ and $4\pi /3<\phi \le 2\pi$, with $\phi =\pi /3, \pi , 5\pi /3$ as the center angle, respectively. Due to the phase tilt, the light field energy in the three sectors will converge towards the three center angles. The spiral phase term factor makes it so that there is a phase singularity on the central axis of the beam to ensure that the light field energy in the three sectors will not converge to one focal spot in the focal plane via the self-focusing. Therefore, we can finally obtain three independent focal spots located at an angle $\phi =\pi /3, \pi , 5\pi /3$, respectively, in the focal plane. To confirm this analysis, we will calculate the light intensity distribution of the modulated CAB during the propagation according Equation (4).

From Equations (3) and (4), we can calculate the light intensity distribution of the modulated CAB during the propagation (Figure 2). The simulation parameters used in the calculation are $\lambda =532 \mathrm{nm},$ $r_0=1 \mathrm{mm}, a=0.1, w=0.088$. Figure 2a is the normalized light intensity distribution in the xz plane, from which we can see that the CAB carrying azimuthally spliced power-exponential phase still retains the self-focusing property. Figure 2(b1–b6) are, respectively, the normalized light intensity distributions and phase distributions of the beam at different propagation planes, whose locations are shown by the white-dotted lines in Figure 2a. Clearly, the beam has a circularly symmetric input intensity pattern [see Figure 2(b1)]. Meanwhile, during the propagation, before the focus, the intensity rings will be gradually divided into three parts, which converge toward the center (see Figure 2(b2–b4)). When the beam propagates to the focal plane (see Figure 2(b5)), three independent focal spots are obtained as expected. Then, after the focus plane, as shown in Figure 2(b6), the beam will gradually diverge. The results of Figure 2 show that our proposed modulation method for focal spot shaping of CAB is effective.

To further confirm the effectiveness of our modulation method, an experiment was performed. The schematic diagram of the experimental setup for generating the CAB carrying azimuthally spliced power-exponential phase is depicted in Figure 3. A linearly polarized Gaussian beam is generated by a semiconductor laser (MLL-F-532nm-2W-EF60784, Changchun New Industries, Changchun, Chian) with a wavelength of $\lambda =532 \mathrm{nm}$. To eliminate the inhomogeneity of the beam for improving its quality, the beam is incident into a spatial filtering system which mainly consists of two lenses and precise pinholes. The beam is expanded using a beam expander mirror, and then, a line polarizer and a half-wave plate are utilized to control the power of the beam and to match the polarization requirements of the spatial light modulator (SLM); then, the beam will be reflected by a phased SLM (HOLOEYE PLUTO-2.1, 1920 × 1080 pixels, 8 μm, Holoeye, Berlin, Germany), which is uploaded with predesigned holograms that contain the formation of the proposed beams. Similarly to Equation (6) in ref. [20], we use the interference pattern of the CAB carrying azimuthally spliced power-exponential phase and a plane wave, i.e., ${\Phi }_{SLM}=|1+E\left(r,\phi ,0\right)\exp (2i\pi {\mathrm{f}}_{\mathrm{x}}x){|}^2$ as the holograms. Here, f_x stands for the grating frequency. In our experiment, we set f_x = 50 mm⁻¹. Then, the appropriate fringes, namely, the positive first-order fringe of the propagating beams on the spectral plane, will be selected to undergo the 4f filter system, which includes two thin lenses (f1 = f2 = 150 mm) and a circular aperture. The beam that passes through the 4f filter system and enters a CCD camera (MER2-160-227U3M, 1440 × 1080 pixels, DAHENG IMAGING, Beijing, Chian) for easy observation (consider the back focal plane of L2 as z = 0).

The corresponding experimental results demonstrate the generation of the CAB carrying azimuthally spliced power-exponential phase in free space, as shown in Figure 4. All the parameters are the same as those in Figure 2. We find that the CAB carrying azimuthally spliced power-exponential phase still autofocuses in space and forms three mutually independent focal spots in the focal plane (z = 640 mm). Comparing the measured cross-sectional intensity profile, it is apparent to find that the experimental results align well with the numerical simulation shown in Figure 2. To further investigate the characteristics of the CAB carrying azimuthally spliced power-exponential phase, we present measurements obtained using the same modulation to produce varying numbers of focal spots (in this case, the topological charge in the spiral phase factor l = 2). The total number of main lobes is shown in Figure 5. To illustrate the feasibility of increasing the number of focal spots, we also used the same method to generate six focal spots. As can be seen from Figure 6(a1), the focal spots are not well separated. However, when we set l = 3, the radius of the main ring increases accordingly, and the six focal points can be well separated (see Figure 6(a2)). This shows that we can obtain more focal spots by adjusting the parameters appropriately, which also shows that our method is feasible. (It is reasonable to anticipate minor discrepancies between the experimental and simulated results due to force majeure factors.

3. Evolution of the Modulated CAB in Tight Focus and Its Trapping-Force Properties

We discuss particle-trapping properties of the angular splicing power exponent phase CAB. Therefore, an objective lens with focal length f = 1 mm and NA = 0.95 is considered to focus the modulated CAB. At the same time, due to the spin-orbital coupling effect in tight focusing, after the linear polarized beam is focused by the objective lens, the left-handed circular polarization and right-handed circular polarization components will produce different-order orbital angular momentum beams, respectively. And the coherent superposition of the different-order orbital angular momentum beams will affect the optical field distribution in the focal plane, forming a complex intensity pattern. So, when we talk about trapping properties, we consider that the polarization of the light field is a single circular polarization state. Experimentally, it can be achieved, passing the CAB carrying azimuthally spliced power-exponential phase generated by SLM through a circular polarizer. Considering that the polarization state of the modulated CAB is right-handed circular polarization, based on Richards–Wolf theory, the initial plane of the modulated CAB is taken as the input light field, and its electric field distribution and magnetic field distribution can be expressed as follows [27]:

$$\begin{array}{cc}\mathit{E}\left({\rho }_p,{\varphi }_p,z_p\right)\hfill & ={\displaystyle \frac{-ik\mathrm{f}}{2\pi }}{\int }_0^{2\pi }{\int }_0^{{\theta }_{\max }}E(r,\phi ,0)\sqrt{\cos \theta }\sin \theta \exp \left\{ik[-{\rho }_P\sin \theta \cos \left(\phi -{\varphi }_P\right)+z_P\cos \theta ]\right\}\hfill \\ & \times \left\{\begin{array}{l}\{[{\sin }^2\phi +{\cos }^2\phi \cos \theta ]\\ +[i(\cos \theta -1)\sin \phi \cos \phi ]\}{\mathit{e}}_{\mathit{x}}\\ \{[(\cos \theta -1)\sin \phi \cos \phi ]\\ +[i({\cos }^2\phi +\cos \theta {\sin }^2\phi ]\}{\mathit{e}}_{\mathit{y}}\\ \{[\sin \theta \cos \phi ]+i[\sin \theta \sin \phi ]\}{\mathit{e}}_{\mathit{z}}\end{array}\right\}\mathrm{d}\theta \mathrm{d}\phi ,\hfill \\ \mathit{H}\left({\rho }_P,{\varphi }_P,z_P\right)\hfill & ={\displaystyle \frac{-ik\mathrm{f}\sqrt{\frac{\epsilon }{\mu }}}{2\pi }}{{\displaystyle \int }}_0^{2\pi }{{\displaystyle \int }}_0^{{\theta }_{\max }}E(r,\phi ,0)\sqrt{\cos \theta }\sin \theta \exp \left\{ik\left[-{\rho }_P\sin \theta \cos \left(\phi -{\varphi }_P\right)+z_P\cos \theta \right]\right\}\hfill \\ & \times \left\{\begin{array}{l}\{[(\cos \theta -1)\sin \phi \cos \phi ]\\ -i[{\sin }^2\phi +\cos \theta {\cos }^2\phi ]\}{\mathit{e}}_{\mathit{x}}\\ [{\cos }^2\phi +\cos \theta {\sin }^2\phi ]\\ -i[(\cos \theta -1)\sin \phi \cos \phi ]{\mathit{e}}_{\mathit{y}}\\ \{[\sin \theta \sin \phi ]-i[\sin \theta \cos \phi ]\}{\mathit{e}}_{\mathit{z}}\end{array}\right\}\mathrm{d}\theta \mathrm{d}\phi ,\hfill \end{array}$$

(5)

by using ${\displaystyle \underset{0}{\overset{2\pi }{\int }}\cos n\varphi e^{i\rho \cos (\varphi -\gamma )}d\varphi }=2\pi i^n{J}_n(\rho )\cos n\gamma ,$ ${\displaystyle \underset{0}{\overset{2\pi }{\int }}\sin n\varphi }$ $e^{i\rho \cos (\varphi -\gamma )}d\varphi =2\pi i^n{J}_n(\rho )\sin n\gamma$ and Euler’s formula $e^{ix}=\cos x+i\sin x$; here, $J_n(\rho )$ is the Bessel function of the first kind and order n, and Equation (5) can be reduced to the following:

$$\begin{array}{l}\mathit{E}\left({\rho }_p,{\varphi }_p,z_p\right)={\displaystyle \frac{-ik\mathrm{f}}{2\pi }}{\int }_0^{{\theta }_{\max }}E(r,\phi ,0)\sqrt{\cos \theta }\exp (ik{z}_P\cos \theta )\left\{\begin{array}{l}[0.5(\cos \theta +1)I_l+0.5(\cos \theta -1)I_{l+2}]{\mathit{e}}_{\mathit{x}}\\ [0.5i(1-\cos \theta )I_{l+2}+0.5(\cos \theta +1)I_l]{\mathit{e}}_{\mathit{y}}\\ (\sin \theta I_{l+1}){\mathit{e}}_{\mathit{z}}\end{array}\right\}\sin \theta d\theta ,\\ \mathit{H}\left({\rho }_p,{\varphi }_p,z_p\right)={\displaystyle \frac{-ik\mathrm{f}\sqrt{\frac{\epsilon }{\mu }}}{2\pi }}{\int }_0^{{\theta }_{\max }}E(r,\phi ,0)\sqrt{\cos \theta }\exp (ik{z}_P\cos \theta )\left\{\begin{array}{l}[0.5i(\cos \theta +1)I_l+0.5i(\cos \theta -1)I_{l-2}]{\mathit{e}}_{\mathit{x}}\\ [0.5(\cos \theta +1)I_l+0.5(1-\cos \theta )I_{l+2}]{\mathit{e}}_{\mathit{y}}\\ (-i\sin \theta I_{l+1}){\mathit{e}}_z\end{array}\right\}\sin \theta d\theta ,\end{array}$$

(6)

where

$$\begin{array}{l}{I}_l=2\pi i^l{J}_l(-k{\rho }_p\sin \theta )e^{il{\varphi }_p}\\ I_{l-2}=2\pi i^{(l-2)}{J}_{(l-2)}(-k{\rho }_p\sin \theta )e^{i(l-2){\varphi }_p}\\ I_{l+2}=2\pi i^{(l+2)}{J}_{(l+2)}(-k{\rho }_p\sin \theta )e^{i(l+2){\varphi }_p}\\ I_{l+1}=2\pi i^{(l+1)}{J}_{(l+2)}(-k{\rho }_p\sin \theta )e^{i(l+1){\varphi }_p},\end{array}$$

(7)

where $\left({\rho }_p,{\varphi }_p,z_p\right)$ represent spherical coordinates, ${\mathit{e}}_{\mathit{x}},{\mathit{e}}_{\mathit{y}}$, and ${\mathit{e}}_{\mathit{z}}$ are the unit vectors along the x, y, and z directions, respectively, $\theta$ and $\phi$ denote the polar and azimuthal angles in polar coordinates, and ${\theta }_{\max }=\arcsin (NA)$ denotes the semi-aperture angle associated with the objective lens. The focal length satisfies the sine condition $r=\mathrm{f}\sin \theta$, and $\epsilon$ and $\mu$ are the permittivity and magnetic permeability in vacuum, respectively.

Figure 7(a1–a4) depict the intensity distribution and phase (illustration) of different polarization electric field components and the total intensity distribution of CAB carrying azimuthally spliced power-exponential phase in the tight focusing focal plane. It can be seen that when the CAB carrying azimuthally spliced power-exponential phase is focused by a lens with a high numerical aperture, the focal plane light field distribution with three focal spots is obtained.

Under the same parameters, we further calculate the energy flux density distribution $\mathit{S}={\displaystyle \frac{1}{2\mu }}Re(\mathit{E}\times {\mathit{H}}^{*})$ and spin angular momentum density distribution $J_{SAM}={\displaystyle \frac{\epsilon \mathrm{Im}({\mathit{E}}^{*}\times \mathit{E})}{4\omega }}$ near the focal plane [28], where $\omega =kc$ (c is the speed of light in a vacuum), and * represents the complex conjugation of the variables. Both the particle and the surrounding medium are commonly nonmagnetic. The particle predominantly reacts to the electric part of the focused field. Hence, only the electric SAM density is considered here.

Figure 8(a1–a3) are the cross-sectional energy flux density of the modulated CAB in front and behind the focal plane, where Figure 8(a1–a3) correspond to the plane in front of the focal plane $5\lambda$, the focal plane, and the plane behind the focal plane $5\lambda$, respectively. It can be seen that in front of focal plane, the light field energy converges toward the center of the three regions (see Figure 8(a1)), and the maximum flow can be reached at the focal plane (see Figure 8(a2)); then, after the beam continues to propagate through the focal plane, the light field energy gradually diffuses outward (see Figure 8(a3)). In the above transmission process, due to the existence of the spiral phase, the optical field energy flux density has an angular component, indicating that the optical field energy flows in the angular direction under the action of spiral phase. Figure 8(a4) shows the cross-sectional spin density vector distribution of the tuned CAB in the focal plane; in optical manipulation applications, the spin density distribution affects the spin of the captured particle.

Let us further discuss the light force of the particles trapped by the above light field. According to Rayleigh scattering theory, in the light field, Rayleigh particles are subject to gradients and scattering forces, which can be expressed as follows [29]:

$$\begin{array}{l}{F}_{grad}={\displaystyle \frac{2\pi n_m{R}^3}{c}}\left({\displaystyle \frac{n^2-1}{n^2+2}}\right)\nabla \tilde{I}(x,y,z)\\ F_{scat}={\displaystyle \frac{8\pi n_m{k}^4{R}^6}{3c}}{\left({\displaystyle \frac{n^2-1}{n^2+2}}\right)}^2\tilde{I}(x,y,z)\widehat{\mathit{z}},\end{array}$$

(8)

where R represents the radius of Rayleigh particles, and $n=n_p/n_m$ is the relative refractive index of the particles for the medium, where $n_m$ is the refractive index of the medium, and $n_p$ is the refractive index of the particles. The polystyrene spheres with a refractive index of 1.59 and a radius of 50 nm were selected for the calculations of the present study, and an aqueous solution was selected for the medium. $\tilde{I}(x,y,z)$ is the average light intensity on the plane and can be expressed as $\tilde{I}(x,y,z)=n_m{\epsilon }_{0}c{\left|\mathit{E}\left({\rho }_p,{\varphi }_p,z_p\right)\right|}^2/2$.

According to Equation (8), the optical force near the focal plane is calculated as shown in Figure 9, where Figure 9(a1) is the transverse gradient force, Figure 9(a2) is the distribution of longitudinal scattering force in cross-section, Figure 9(b1) is the longitudinal gradient force and scattering force, and Figure 9(b2) is the longitudinal total force (when calculating the longitudinal force, we chose the center axis position of the leftmost focal point of one of the three focal points.). From Figure 9(a1), it can be seen that the focal plane light field can form three transverse gradient force potential wells, the center of which is located at the center of the three focal points. In addition, from Figure 9(a1–a3), it can be seen that the light-scattering force on the focal axis is two orders of magnitude smaller than the gradient force, so three stable 3D stable-capture potential wells can be obtained at the focal plane.

In the above, we have proven through theoretical simulation and experiment that four and five focal points can also be effectively obtained in the focal plane through controlling the number of segments of the spliced power-exponential phase. Therefore, we further discussed the optical force near the focal plane of the light field in these two cases. Under the same calculation parameters, the simulation calculation results are shown in Figure 10, where Figure 10(a1–a3) are transverse gradient force, distribution of longitudinal scattering force in cross-section, and longitudinal total force, respectively, when forming four focal points; Figure 10(b1–b3) are transverse gradient force, distribution of longitudinal scattering force in cross-section, and longitudinal total force, respectively, when forming five focal points. It is not difficult to see from the results in Figure 10 that the CAB of four and five segments of power-exponential phase can obtain four and five stable 3D capture potential wells in the focal plane, respectively.

4. Conclusions

In conclusion, we theoretically propose and experimentally prove a method to regulate the number of focal points in the CAB focal plane by the azimuthally spliced power-exponential phase. The results show that this method can effectively obtain multiple independent focal points in the focal plane without chaotic background fields. Through the simulation of light forces near the focal plane, it is found that these focal points can generate multiple stable 3D trapping potential wells, which can be used to achieve multi-particle trapping. The research in this paper provides a new insight for the focal spot shaping of CAB light field and has potential application value in the fields of intense light-bullet wave packets generation, laser processing, and optical manipulation related to CAB.