Quantitative characterization of autofocusing and trapping of multi-Airy vortex beams

Abruptly autofocusing vortex beams were proposed for a variety of applications such as optical manipulation, but their quantitative characterization of their trapping capability remains largely unexplored. In this paper, we propose a type of autofocusing vortex beams named multi-Airy vortex beams (MAVBs) and investigate their autofocusing and trapping characteristics theoretically and experimentally. It is found that the autofocusing performance and trapping ability of MAVBs can be enhanced by adjusting the relevant parameters and increasing the number of superposed Airy within a certain range. Although the trapping force and trapping stiffness of MAVBs decrease as the topological charge increases, we found that MAVBs have advantages in capturing and manipulating large-sized particles in our experiments. In the process of manipulating multiple particles for rotation, the larger the topological charge, the more particles can be rotated simultaneously by MAVBs. Moreover, the speed and even the direction of rotation can be easily adjusted by controlling the degree of dispersion between particles. Our work provides a guide to quantitatively exploring the optical trapping capabilities of autofocusing vortex beams, which may lead to different photonic tools for optical trapping and manipulation.

Figure 1

Propagation of MAVBs ($l=1$) with different superimposed Airy numbers $N$ ($P=1W$). (a) Experimental setup for generating MAVBs: P1 and P2, polarizers; SLM, spatial light modulator; L, lenses; I, irises; CCD, charge-coupled device. (b) $N=4$; (c) $N=8$. (b1) and (c1) and (b2) and (c2) are the intensity distributions of MAVBs ($l=1$, $N=4,8$) in the initial position, and the focal positions $f=98\text{mm}$ ($N=4$) and 96 mm ($N=8$), respectively. (b3) and (c3) are the side view. The white curve in the middle is the corresponding intensity distribution curve, and the illustration in the upper right corner shows the experimental results. The details of the transmission process can be viewed in Supplemental Videos 1–2 [26].

Figure 2

Propagation of MAVBs ($N=12$) with different TCs. (a) $l=1$; (b) $l=2$; (c) $l=3$. (a1)–(c1) and (a2)–(c2) are the intensity distribution of MAVBs ($N=12$, $l=1,2,3$) in the initial position, and the focal positions $f=96\text{mm}$ ($l=1$), 95 mm ($l=2$), 94 mm ($l=3$), respectively. (a3)–(c3) are the side view. The white curve in the middle is the corresponding intensity distribution curve, and the illustration in the upper right corner shows the experimental results. The details of the transmission process can be viewed in Supplemental Videos 3–5 [26].

Figure 3

Autofocusing performance of MAVBs ($N=12$) with different parameters. (a)–(d) are the changes of autofocusing coefficient $K(I_{\text{max}}/I_0)$ corresponding to $N$ of MAVBs with different TCs ($l=0,1,2,3$), exponential decay factor ($\alpha =0.02,0.04,0.06$), arbitrary transverse scale factor ($r_0=40,50,60\mu \mathrm{m}$), and transverse offset ($c_0=80,90,100\mu \mathrm{m}$), respectively.

Figure 4

The transversal power flow of MAVBs ($N=12$) in the focal position, where the black arrows represent the direction of energy flow. (a) $l=1$; (b) $l=2$; (c) $l=3$. See Videos 6–8 in the Supplemental Material [26].

Figure 5

(a) Experimental setup for trapping and manipulating particles. Laser, $\lambda =1064\text{nm}$; PBS, polarizing beam splitter; BE, beam expanders; M, mirror; SLM, spatial light modulator; L, lenses; BS, beam splitter; CCD, charge-coupled device; Objective, oil objective [numerical aperture $\left(\mathrm{NA}\right)=1.25,\times 100]$; QPD, quadrant photoelectric detectors. (b) Phase image of MAVBs. BG, blazed grating.

Figure 6

The ability of MAVBs to capture single PS microparticles. (a1)–(c1) Experiment: Power spectral density (PSD) maps of single PS ($3\mu \mathrm{m}$ red; $4\mu \mathrm{m}$ yellow) captured by MAVBs with different topological charges ($l=1,2,3$) at a power of $P=32\text{mW}$. The dashed lines indicate the position of the $f_c$, and the trapping stiffness ${\kappa }_r$ (${\kappa }_r=f_c*2\pi \gamma$) is shown in the lower left-hand corner. (a2)–(c2) Simulation: The trapping force distribution of MAVBs. The bottom left panel displays the corresponding trapping stiffness ${\kappa }_r$ (${\kappa }_r=dF/dr$).

Figure 7

MAVBs ($N=12$, $l=1$) rotate multiple PS ($D_s=1.5\mu \mathrm{m}$) particles. (a)–(g) MAVBs manipulate five PS particles to rotate along the clockwise direction. (h)–(n) After adjusting the degree of dispersion between particles, the direction of the rotation of particles changes from clockwise to counterclockwise. See Videos 12–14 in the Supplemental Material [26].