Massive Parallel Sorting of Particles Using Unwound Polygonal Vortex Beams

1 Introduction

For decades, optical tweezers have been used to sort cells in a fluid environment [1, 2]. Tweezers use a single focused beam to trap and manipulate high-refractive-index particles [3]; the technique was a natural evolution of the early work of Ashkin using dual beams for trapping [4]. Optical tweezers are an ideal tool for micromanipulation because they are non-contact and non-destructive; therefore, they are widely used in life sciences [5], biology [6], biophysics [7], and colloidal sciences [8, 9].

In practice, many varieties of optical tweezers have been realized [10–16], such as plasmonic tweezers [10], non-linear tweezers [11], vortex tweezers [12, 13], optical trap arrays [14–16], and more [17–21]. But single function tweezers designed to trap an object cannot satisfy all desired tasks [12, 22–28], such as the transportation [12], micromachining [22], and sorting [23–27] of multiple particles.

Mixed particles in a background environment can be separated by optical sorting [1, 2, 29]. Optical sorting selects target particles from a mixed distribution of particles using their differing responses to optical forces [16, 23–27, 30, 31]. Microscopic particles are sorted by an optical lattice, which has a high sorting efficiency [16]. Living and dead cells have been massively parallel separated by optical imaging, with live cells trapped in an optical ring [24]. Yeast cells of different sizes have been effectively separated and transported by dual-channel line optical tweezers [30]. The Junin virus particles were successfully sorted based on the light scattering parameters of fluorescent channels, which also can compare particles secreted by virus and human primary macrophages [27]. Nanoscale particles with sizes from 20 to 110 nm have been separated by nanoscale deterministic lateral displacement arrays [25]. Soft particles can be sorted from rigid particles using a microfluidic device, which can realize real-time one-by-one sorting and a sorting purity of 88% [26]. Although some devices have been made that can enhance the sorting efficiency [23, 24, 30, 31], it is often not enough for desired applications.

Beams with vortex structures have been shown to be quite versatile in trapping applications, both in the trapping of low-index particles [32] and in the rotation of particles [33]. Optical vortex (OV) beams have an azimuthal phase factor exp(ilθ), where θ denotes the azimuth angle and l is an integer known as the topological charge. Traditional vortex beams are symmetric around their central axis [34–37]; modulated vortex beams with a specific polygon shape can be generated by manipulating the vortex phase function, and the utility of such beams in optical manipulation has been demonstrated experimentally [38, 39]. An OV beam can be transformed into a linear intensity distribution by a geometric transformation, and this is now a standard method to spatially sort beams with different topological charges [40]. Laguerre–Gaussian (LG) beams are the archetypical class of OV beams [41].

In this paper, we modulate an LG beam to obtain a linear OV beam with a kinked distribution to realize massively parallel sorting of particles. The kinked distribution increases the size of the region in which the particles interact with light, significantly improving the sorting capabilities of the system.

2 Methods

The field of an OV beam of radial order n and azimuthal order l in the waist plane may be written as

$U\left(\rho ,\theta \right)=A_{nl}\left(\rho \right)\exp \left(il\theta \right),(1)$

where (ρ, θ) are polar coordinates, A_nl(ρ) represents the amplitude of the beam, and exp (ilθ) is the vortex phase term with topological charge l [41]. The intensity distribution is the number of circular rings equal to the order n + 1.

If one replaces the vortex phase with a term exp[iϕ(θ)], then the shape of the OV beam is determined by the function ϕ(θ) [38]. A polygon OV beam can be generated in the far field by manipulating the function ϕ(θ) of an LG beam with radial order n = 0. Then, the field can be expressed as follows:

$U\left(\rho ,\theta \right)=\frac{A_0}{w}{\left(\frac{\sqrt{2}\rho }{w}\right)}^{\left|l\right|}\exp \left(-\frac{{\rho }^2}{w^2}\right)\exp \left(i\varphi \left(\theta \right)\right),(2)$

where A₀ is a constant and w denotes the beam width. The phase ϕ(θ) is written as

$\varphi \left(\theta \right)=2\pi \left(l+b\right)\frac{\underset{0}{\overset{\theta }{\int }}R\left(\phi \right)d\phi }{\underset{0}{\overset{2\pi }{\int }}R\left(\phi \right)d\phi }-b\theta ,(3)$

with

$R\left(\phi \right)=\left\{\begin{array}{c}\frac{r_2{r}_1\sin \left({\phi }_1-{\phi }_2\right)}{r_2\sin \left(\phi -{\phi }_2\right)-r_1\sin \left(\phi -{\phi }_1\right)}for{\phi }_1=0\le \phi \le {\phi }_2\hfill \\ \hfill ⋮\hfill \\ \frac{r_{i+1}{r}_i\sin \left({\phi }_i-{\phi }_{i+1}\right)}{r_{i+1}\sin \left(\phi -{\phi }_{i+1}\right)-r_i\sin \left(\phi -{\phi }_i\right)}for{\phi }_i\le \phi \le {\phi }_{i+1}\hfill \\ \hfill ⋮\hfill \\ \frac{r_n{r}_1\sin \left({\phi }_n-2\pi \right)}{r_1\sin \left(\phi -2\pi \right)-r_n\sin \left(\phi -{\phi }_n\right)}for{\phi }_n\le \phi \le 2\pi ,\hfill \end{array}\right.,(4)$

where b is a constant, R(φ) is the mathematical expression for the side of a convex polygon, and (r_i, φ_i) represents the polar coordinates of the vertices of the polygon. For simplicity, the starting vertex (r₁, φ₁) is fixed at φ = 0. The geometric distribution of the intensity can be designed by modulating the piecewise function R(φ).

By sending a polygon OV beam through the aforementioned geometric transformer, we can unwind it into a linear OV beam; the corners of the polygon are mapped into kinks in the linear beam. The transformation phase function is [40, 42]

$\mathrm{\Psi }\left(x,y\right)=\frac{2\pi a_1}{\lambda f}\left[y\arctan \left(\frac{y}{x}\right)-x\ln \left(\frac{\sqrt{x^2+y^2}}{{\mathrm{a}}_2}\right)+x\right],(5)$

and the output field is

$U\left(u,v\right)=\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}U\left(x,y\right)\exp \left[i\mathrm{\Psi }\left(x,y\right)\right]\times \exp \left[-i2\pi \left(ux+vy\right)\right]dxdy,(6)$

where (x, y), (u, v) are the Cartesian coordinates in the polygon OV field and the Fourier plane, respectively. U(u, v) is the electric field of the linear OV beam with a kinked distribution, and f denotes the focal length of the lens. Here, a₁ and a₂ are independent constants which determine the size and the position of the transformed field. Figure 1 displays the intensity distributions of the input and linear OV beams with parameters f = 50mm, w = 0.1mm, l = 25, b = −0.4 × l, 2πa₁ = 1mm, and a₂ = 4.5 mm.

FIGURE 1

FIGURE 1. Intensity distribution of polygon OV beams: (A) circular, (B) triangle, and (C) pentagon. (D–F) Linear OV beams with kinked distribution corresponding to (A–C).

Applying Eqs 2–4, the shapes of polygon OV beams are chosen as circular, triangular, and pentagonal (see Figures 1A–C); the corresponding unwound intensity distributions of the linear OV beams can be seen in Figures 1D–F. They all have the same extension h in the v direction. Figure 1D is a beam without kinks; Figure 1E and Figure 1F are beams with the intensity distribution carrying kinks. It is obvious that the kinks are related to the shape of the polygonal OV beams. We use these beams to enhance the efficiency in particle sorting.

Based on the fractal theory, the kinked profile of the intensity pattern is a non-rectifiable curve [43, 44]. Cut the curve into N integer segments (N > 1), and the fractal dimension D can be expressed as D = − ln N/ln r(N), where r(N) is the similarity ratio [43]. The length of the curve for Figure 1 is defined as L = hN^1−1/D. D will be more accurate with the smaller value of r(N), and the length L will be longer. In Figures 1D–F, the fractal dimension D is determined by the kinks (see Figure 1E and Figure 1F). The linear OV beam with kinked distribution satisfies D > 1; as a result, L is larger than h. To the same extent h, the more the kinks are, the longer the L will be. Therefore, the length of the pattern in Figure 1F is the longest compared to that in Figure 1D and Figure 1E.

Because of the kinks, the length of the intensity pattern is longer than the spatial extent h of the beam, increasing the region over which the beam can interact with particles. This enhancement can be used in particle sorting to enhance the sorting efficiency.

In the sorting platform, two kinds of particles with different refractive indices are mixed in a fluid environment. For example, the power $A_0^2=0.1$ W/um² [13], the particles have the same sphere radius a = 10nm, and the refractive index of fluid media is n_m = 1.33 (water). The refractive indices of particles are n_ph = 1.59 (red), n_pl = 1 (blue), respectively. There are 180 spheres in total, and each color has 90 spheres. In Figure 2, the sorting platform is put in the focal plane of the lens. When the beam is fixed, we move the platform, and the particles with a higher refractive index can be arranged to move with the beam stably. Thus, two kinds of particles can be sorted. Figure 2 shows how the unwound circular and polygonal OV beams are used to sort particles. In Figure 2A, 26 red spheres are missed by the sorting process (see Supplementary Movie S1). The sorting efficiency of this beam approaches 71.1%. In Figure 2B, there are 10 particles left unsorted (seeSupplementary Movie S2) with an increasing efficiency of 88.9%. In Figure 2C, all the particles are sorted successfully (see Supplementary Movie S3). In this case, the sorting efficiency reaches 100%. From Figure 2, one can see that the unwound polygonal OV beams sort more particles compared to the unwound circular OV beam. The sorting efficiency is increased nearly 18% (see Figure 2A and Figure 2B). Comparing Figure 2B with Figure 2C, one can see that more kinks lead the sorting efficiency rise to 100% from 88.9%. As a result, there is a higher efficiency for particle sorting when increasing the length of the linear OV beam. Therefore, this setup can realize massively parallel sorting of particles.

FIGURE 2

FIGURE 2. Particles with a higher refractive index (red spheres) and lower refractive index (blue spheres) are sorted by the optical sorting platform. A linear OV beam (A) without kinks (see Supplementary Movie S1), (B) with kinks corresponding to a triangle (see Supplementary Movie S2), and (C) with kinks corresponding to a pentagon (see Supplementary Movie S3). (D) Optical lattice (see Supplementary Movie S4).

To analyze the sorting capacity of the generated beam, we consider the force acting on the particles. The radiation force consists of the scattering force F_scat and the gradient force F_grad. According to the theory of Rayleigh scattering, when the radius of a spherical nanoparticle a ≪ λ, the two kinds of forces can be calculated separately [45]. In order to stably manipulate particles, F_grad is supposed to be large enough to overcome F_scat. The buoyancy, gravity, and Brownian forces can be neglected by decreasing temperature and increasing light intensity [46]. The expressions of the gradient force F_grad and scatter force F_scat are

$F_{grad}=2\pi n_m\beta \nabla I/c,\beta =a^3\left(m^2-1\right)/\left(m^2+2\right),(7)$

$F_{scat}=n_m\alpha I/c,\alpha =\left(8/3\right)\pi {\left(ka\right)}^4{a}^2\times {\left[\left(m^2-1\right)/\left(m^2+2\right)\right]}^2,(8)$

where I = |U(u, v)|² is the intensity of the focused beam, a represents the particle size, c is the light speed, and m = n_p/n_m, in which n_m, n_p are the refractive indices of the environment and the Rayleigh particles, respectively. We calculate the gradient force and the scatter force in the horizontal direction (v = 0). Figure 3 shows the radiation forces of the particles in a linear OV beam without kinks (see Figure 3A₁, Figure 3B₁), with kinks corresponding to a triangle (see Figure 3A₂, Figure 3B₂) and a pentagon (see Figure 3A₃, Figure 3B₃). The red and blue curves represent the radiation force of particles with higher and lower refractive indices, respectively. Comparing the three cases, as the kinks increase, the maximum value of the radiation forces is slightly different. By comparing Figure 3A with Figure 3B, it is obvious that the gradient force F_grad is about 10 orders of magnitude larger than the scatter force F_scat in each case. From Figure 3B, one can see that the direction of the gradient force F_grad of the two kinds of particles is different. The sign of the gradient force F_grad denotes the direction of the force. The positive F_grad represents that the gradient force is along the positive u direction, and the negative value represents that the force is along the negative u direction. In Figure 3B, there is only one stable equilibrium point for the particles with high refractive index, which is marked by a solid black dot. Therefore, only the particles with a higher refractive index can be stably manipulated. This theoretical analysis demonstrates that only the particles with a higher refractive index can move with the unwound polygonal OV beam.

FIGURE 3

FIGURE 3. Radiation forces of the two kinds of particles in linear OV beams. The blue dashed lines and the red solid lines represent the particles with low and high refractive indices, respectively. (A₁–A₃) Scattering force with different numbers of kinks. (B₁–B₃) Gradient force with different numbers of kinks.

We give the simulation analysis of the microfluidic sorting system [16]. We use the Gaussian beam with the beam waist size w = 0.1 mm, the power $A_0^2=0.1$ W/um², and the focal lens f = 50 mm. In the back focal plane, the Gaussian lattices with different numbers of arrays are shown in Figure 4. It is shown that the array is not infinite for the same beam size because the lattice structure is disappearing when the number of arrays is increasing. We analyze the radiation forces of particles with higher refractive indices in Gaussian lattices with different numbers of arrays. We define $m=\frac{F_{grad}}{F_{scat}}$ to show the order of magnitude between the gradient force F_grad and the scatter force F_scat. According to the theory of Rayleigh scattering, the particles can be stably trapped when the gradient force F_grad is about one orders of magnitude larger than the scattering force F_scat. The optimum value of m is nearly 10. In Figure 5, we display the order of magnitude m with different numbers of arrays. It is shown that as the number of arrays increases, the order of magnitude m is decreasing and gradually less than 1. It is demonstrated that the particles cannot be trapped when there are too much arrays in a finite space. It means the number of arrays cannot be infinitely increasing. Therefore, the sorting lattice is finite in a finite space.

FIGURE 4

FIGURE 4. Intensity distribution of the Gaussian lattice with the same beam size and different numbers of arrays. (A) 35×35 array; (B) 45×45 array; (C) 55×55 array; (D) 60×60 array.

FIGURE 5

FIGURE 5. Order of magnitude m with different numbers of Gaussian lattices n.

From the above analysis, it is obvious that the space interval of the adjacent array needs to be sufficient to stably trap particles. Therefore, we simulate the Gaussian lattice with different arrays when the order of magnitude m is about 10 (see Figure 6). One can see that the beam size increases with the increasing number of arrays. In Ref. [16], the sorting path of particles combines the length of the sorting platform and the beam size. Therefore, the beam size plays an important role in sorting time. The larger the beam size, the more the sorting time. In Figure 7, we display the sorting time in an optical lattice with different numbers of arrays when the speed is 2 mm/s and the length of the platform is 5 mm. It is obvious that more arrays bring more sorting time. In contrast, in our strategy, the sorting path is just the length of the sorting platform. Thus, we added the blue dashed line to represent the sorting time in our strategy. It is obvious that our strategy can save sorting time of particles. In Supplementary Movie S4, it can be seen that the sorting efficiency of the lattice is much less than that of the unwound OV beams (see Figure 2D). In other words, we can sort more particles in the same time than the strategy proposed by M. P. MacDonald.

FIGURE 6

FIGURE 6. Intensity distribution of the Gaussian lattice with different numbers of arrays. (A) 35×35 array; (B) 45×45 array; (C) 55×55 array; (D) 60×60 array.

FIGURE 7

FIGURE 7. Sorting time (s) in the optical lattice with different numbers of arrays n.

3 Conclusion

In conclusion, through the use of unwound polygonal OV beams, we propose a new method for massively parallel sorting. Such unwound polygonal OV beams can be generated by a geometric transformation. The sorting area is greatly enlarged, and the sorting efficiency is enhanced by increasing the region in which the light and particles interact. Our results are of significant interest to researchers engaged in high-capacity particle sorting in many disciplines, such as biomedical science and colloid physics.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, and further inquiries can be directed to the corresponding authors.

Author Contributions

YC and YY conceptualized the idea and designed the methodology. XL carried out the methodology and wrote the original draft. XL, HZ, and YG were involved in the formal analysis. GG, YC, and YY polished the manuscript. All authors read and approved the final manuscript.

Funding

This work was supported by the National Key Research and Development Program of China (2019YFA0705000), National Natural Science Foundation of China (NSFC) (Nos. 91750201, 12192254, 11974219, 11974218, 12174227, 91950104), Innovation Group of Jinan (2018GXRC010), Natural Science Foundation of Shandong Provincial (ZR2019MA028), and Local Science and Technology Development Project of the Central Government (YDZX20203700001766).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2022.877804/full#supplementary-material

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