Manipulation force analysis of nanoparticles with ultra-high numerical aperture metalens

Yan Wang; Yan Wang; Yan Wang; Miao Peng; Miao Peng; Wei Cheng; Wei Cheng; Zheng Peng; Zheng Peng; Hao Cheng; Xiaodong Ren; Shengyin Zang; Yubei Shuai; Hao Liu; Jiagui Wu; Jiagui Wu; Junbo Yang; Junbo Yang

doi:10.1364/OE.462869

1. Introduction

Optical tweezers technology is an important research topic in the field of optics. It mainly uses the mechanical effect of optical binding to perform non-contact manipulation on particles, which plays an important role in the fields of biology, physics, and micro-nano optics [1–7]. In 1986, Ashkin's team succeeded in trapping particles through focusing a single-beam laser [8], which gave rise to the rapid development of optical tweezers technology. However, traditional optical tweezers are expensive, bulky, complex in structure, and limited by the diffraction limit since the particles that can be trapped are usually in the micrometer scale [9]. With the rapidly developing metasurfaces manufacturing technology [10–13], more miniaturized and integrated on-chip optical tweezers are being used to trap and manipulate nanoscale particles and smaller molecular biological particles [14].

In recent years, metalenses have demonstrated significant advantages such as highly integrated miniaturization, high flexibility, ultra-high NA, exceeding the diffraction limit, and low loss [15–21]. There are several design methods for metalens, such as Pancharatnam–Berry (PB) phase, transmission phase, and spin Hall effect, which can flexibly generate single focus, multifocal, and vortex optical beams [22–24], making them a favorable tool for realizing on-chip optical tweezers [25]. Metalenses primarily control the phase and amplitude of optical signals by arranging nanopillars in a periodic fashion [26–29], and the resulting focal point in the far-field has a strong gradient force, which enables stable trapping and manipulation of particles and biological cells [30–33]. The polarized optical signal generated by the polarization-sensitive metalens can manipulate liquid crystal droplets to simulate optical motors [34]. It is also possible to use left-handed circularly polarized optical signals and right-handed circularly polarized optical signals to achieve the opposite trap force of the particles, to accurately trap the target particle and simultaneously block other particles [35]. Moreover, the polarization state conversion can also enable drag-and-drop of particles [36]. In addition, the laser emitted from the focal point can produce an optical force on the surface of the metalens, and different pull or thrust transformations can be performed by adjusting the polarization state [37], which is of great significance to aerospace technology.

However, the influence of the wavelengths with bandwidth on the particle trapping stability has not been extensively discussed in the field of metalenses and optical force [38–40]. Owing to the inherent properties of metamaterials, there is chromatic aberration in metalens [41–45], meaning that wavelengths with a certain bandwidth cannot be focused at the same position, leading to a non-constant effect on the optical force, motion state, and trapping position of particles. In biology, constant optical force relative to different wavelengths helps reduce photodamage to biological specimens [46]. In addition, NA is a key characterization of angular range over which the system can accept or emit light. Therefore, high NA of metalens is important to improve the focusing efficiency [47,48], which in turn increases the optical force and trapping efficiency [49,50]. Based on above consideration, we propose a polarization-insensitive achromatic metalens to manipulate and trap SiO₂ particles, demonstrating that its NA can reach 0.97, with an average focusing efficiency of 44%. When using the PSO to optimize the phase of the metalens, the wavelength of 100 nm bandwidth in the range of 1014 nm to 1114 nm achieved an achromatic metalens, which can keep constantly the optical force, motion state, and trapping position of particles, ameliorate the influence of wavelength bandwidth, and increase the practicability of metalenses in the application of optical force.

2. Methods

2.1 Achromatic metalens

The phase distribution formula used for single-focal focusing of the metalens is as follows:

(1)$$\varphi ({x,y,\lambda } )={-} \frac{{2\pi }}{\lambda }\left( {\sqrt {{x^2} + {y^2} + {f^2}} - f} \right)$$

where $({x,y} )$ is the position of the metalens surface from the center, λ is the wavelength, f is the focal length, and $\varphi ({x,y,\lambda } )$ is the phase value. Therefore, focal length varies with wavelength, which is the dispersion of the metalens. We use formula (1) to construct the dispersive metalens. When the focal length remains the same and the wavelength changes, the response phase at the same position also changes. Therefore, it is necessary to perform phase compensation for different wavelengths to realize an achromatic metalens. By introducing the phase factor $C(\lambda )$ and using the particle swarm optimization algorithm to meet the phase compensation required by different wavelengths, the achromatic phase of the metalens can be rewritten as follows:

(2)$${\varphi _A}({x,y,\lambda } )={-} \frac{{2\pi }}{\lambda }\left( {\sqrt {{x^2} + {y^2} + {f^2}} - f} \right) + C(\lambda )$$

Figure 1(a) shows the design of the achromatic metalens optimized by particle swarm optimization, with a radius of 12.5 $\mu $m and consisting of periodically arranged nanopillars and bases. The incident wavelength ranges from 1014 nm to 1114 nm with step 25 nm, the bandwidth is 100 nm, and the focal length is set to 3 µm, meaning the NA of the achromatic metalens can reach 0.97. Figure 1(b) shows a plan view of the achromatic metalens, where the nanopillars vary in radius to satisfy the phase distribution required for the corresponding positions. The nanopillars structures are shown in Fig. 1(c)–(e). The material selected for the base was SiO₂, and the material of the nanopillar was Si, which has a high refractive index in the near-infrared and almost no loss [51]. The period P of the nanopillar was approximately λ/2, and therefore P was set to 500 nm. H is the height of the nanopillar, which was selected to be 650 nm, and R is the radius of the nanopillar, which varied from 50 nm to 200 nm with step 5 nm. The simulation uses the finite-difference time-domain (FDTD) method, and the software is Lumerical FDTD 2016. The incident light sources of metalens and nanopillars are both x polarized plane light sources. For nanopillars and metalens, the simulation boundary in x-direction is anti-symmetric, in y-direction is symmetric, and in z-direction is perfectly matchedlayer (PML). The mesh is set to 5 nm.

Fig. 1. (a) Design diagram of achromatic metalens. (b) Plan view of achromatic metalens. (c-e) Structure of nanopillars.

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In order to meet the phase distribution of achromatic metalens, we used the PSO [52,53] to optimize the phase compensation factor $C(\lambda )$ in Eq. (2). Figure 2 shows a flowchart of the PSO algorithm. First, it needs to generate an initial population size of 50. The population consists of particles, which represent the compensation phase. Each particle has the properties of velocity and position. The velocity determines how fast the particle moves, and the position determines the direction in which the particles move. The velocity and position of each particle also need to be initialized. Next, it needs to calculate the fitness of each particle. The fitness $\Delta f$ is calculated using the difference between the ideal phase and the actual phase.

(3)$$\Delta f = \sum\nolimits_\mathrm{\lambda } {\sum\nolimits_x {\sum\nolimits_y {abs\,({\varphi _{ideal}}({x,y,\mathrm{\lambda }} )- {\varphi _{true}}({x,y,\mathrm{\lambda }} )} } } $$

Here, ${\varphi _{ideal}}$ is the ideal achromatic phase distribution as calculated by Eq. (2), and ${\varphi _{true}}$ is the true phase which is obtained by the FDTD scanning parameters of nanopillars, as shown in Fig. 3(a). In Fig. 3(a), the radii of nanopillars varied from 50 nm to 200 nm, which completely covers 2π across different wavelengths, and satisfies the metalens phase distribution requirements.

Fig. 2. Flow chart of particle swarm optimization algorithm.

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Fig. 3. (a) Phase diagram of nanopillars with radii in the range of 50-200 nm at different wavelengths. (b) Transmission diagram of nanopillar radius in the range of 50-200 nm at different wavelengths. (c) The focal length and FWHM diagram of the achromatic metalens. (d) Focusing efficiency diagram of the achromatic metalens.

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Next, comparing $\Delta f$ to find the optimal solution of particle itself, which is also called the individual extremum (pbest). Then we share the individual extreme value with population, and find the optimal solution in population, which is called the global optimal solution (gbest). And record the positions of individual extrema and the global optimal solution. If the requirements were satisfied, the iteration was ended. Therefore, we choose ${\varphi _{true}}({x,y,\lambda } )$ in $\Delta f$ to construct the achromatic metalens. Otherwise, the velocity and position of each particle were updated based on the pbest and the gbest, where pbest is the position of the individual extreme value, and gbest is the position of the global optimal solution. The algorithm could then enter the next iteration, where the formula is as follows:

(4)$${v_i} = w \times {v_i} + {c_1} \times rand \times ({pbes{t_i} - {x_i}} )+ {c_2} \times rand \times (gbes{t_i} - {x_i})$$

(5)$${x_{i + 1}} = {x_i} + {v_i}$$

where w is the inertia factor which was set to 0.9, ${c_1}$. and ${c_2}$ are the learning factors, which were 1.5 and 0.5, respectively, rand is a random number between (0, 1), v is the particle velocity, and x is the position of particle, analogous to $C(\lambda )$ in Eq. (2).

Figure 3(b) shows the transmission of nanopillars at different wavelengths. The transmission of most nanopillars exceeds 80%. The calculation formula of transmission efficiency is as follows:

(6)$$T({{f_\lambda }}) = \frac{{\frac{1}{2}\smallint Re({P({{f_\lambda }} )} )ds}}{{{P_{in}}}}$$

Here, ${f_\lambda }$ is the frequency of the incident light, $P({{f_\lambda }} )$ is the Poynting vector, s is the surface selected near the top of the nanopillar, and ${P_{in}}$ is the incident optical power. Therefore, the higher the transmission efficiency, the more light is transmitted through the nanopillars. which lays the foundation for the high focusing efficiency of the metalens. Figure 3(c) shows both the focus position and full width at half maxima (FWHM) of the achromatic metalens. The focal length of the achromatic metalens has a gradual change and then remains at 3 µm, and the FWHM increases as the wavelength increases. As shown in Fig. 3(c), the Δλ/λ ($\frac{{\max (f )- \textrm{min}(f )}}{{mean(f )}}$) among 100 nm bandwidth is about 18%, being similar to the Δλ/λ values of Ref. [15]. Furthermore, Fig. 3(d) shows the achromatic metalens focusing efficiency, where the average focusing efficiency is 44%. This focusing efficiency is defined as the ratio of optical intensity within the triple FWHM area to incident optical intensity. The NA and the focusing efficiency of achromatic metalens are compared with following Table 1, which shows a relatively higher NA value and relatively higher focusing efficiency. In addition, high NA and a short focal length (about 3 µm) are beneficial for the subsequent optical force.

Table 1. NA and focus efficiency values compared to other literature

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Figure 4(a-b) shows the normalized intensity distribution of dispersive metalens and achromatic metalens at the axial (x-z) surface, respectively. The focal length of the achromatic metalens is more concentrated than that of the dispersive metalens, which lays the foundations for the constant optical force under different wavelengths. Figure 4(c) shows the normalized intensity distribution of transverse (x-y) surface. The asymmetry of normalized intensity distribution in the x-y plane may be due to the phase modulation of high NA metalens. These two planes are also the optical trapping planes of particles. Figure 4(c) shows the FWHM diagrams of metalens. The FWHM value increases as the wavelength increases. Additionally, high NA and short focal length do affect the achromatic metalens, if considering the constraint relationship among parameters (achromatic, high NA and short focal length) [48,54]. Therefore, the achromatic effect in our case couldn't be as noticeable as that of tens µm long focal length metalens [42,45].

Fig. 4. (a-b) normalized intensity distribution in the axial (x-z) plane of the dispersive metalens and achromatic metalens under different wavelengths, respectively. (c) normalized intensity distribution in the transverse (x-y) plane of the achromatic metalens under different wavelengths. (d) FWHM diagrams of the achromatic metalens under different wavelengths.

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2.2 Optical Force Calculation

We used the integration of Maxwell stress tensor over particle surface s to calculate the optical force in far-field. The Maxwell stress tensor are calculated by electromagnetic numerical calculation obtained from the FDTD. For easy integration, we assume the particle to be enclosed within a cube and calculate the integral of the Maxwell stress tensor on each surface of the cube, in order to represent the optical force on the particle in the x, y, and z directions. The optical force F is as follows:

(7)$$F = \mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc} {{T_{ij}}{{\vec{n}}_j}ds}$$

Here, ${\vec{n}_j}$ is the unit vector normal to the surface, ${T_{ij}}$ is the time-averaged Maxwell stress tensor:

(8)$$\left\langle {{T_{ij}}} \right\rangle = \frac{1}{2}Re\left[ {\varepsilon {E_i}E_j^\ast{+} \mu {H_i}H_j^\ast{-} \frac{1}{2}{\delta_{ij}}({\varepsilon {{|{\vec{E}} |}^2} + \varepsilon {{|{\vec{H}} |}^2}} )} \right]$$

Here, ɛ and u are the relative permittivity and permeability of the simulated environment, respectively. E is the electric field on the surface of the cube, E* is the complex conjugate of the electric field, H is the magnetic field on the surface of the cube, and H* is the complex conjugate of the magnetic field. i and j represent the surfaces of the cube in the x, y, and z directions. ${\delta _{ij}}$ is Kronecker delta. Therefore, ${T_{ij}}$ can be expanded as follows:

(9)$${\displaystyle{T_{ij}} = \left( {\begin{array}{ccc} {\varepsilon E_x^2 + \mu H_x^2 - \frac{1}{2}({\varepsilon {{|{\vec{E}} |}^2} + \mu {{|{\vec{H}} |}^2}} )}&{\varepsilon {E_x}{E_y} + \mu {H_x}{H_y}}&{\varepsilon {E_x}{E_z} + \mu {H_x}{H_z}}\\ {\varepsilon {E_x}{E_y} + \mu {H_x}{H_y}}&{\varepsilon E_y^2 + \mu H_y^2 - \frac{1}{2}({\varepsilon {{|{\vec{E}} |}^2} + \mu {{|{\vec{H}} |}^2}} )}&{\varepsilon {E_y}{E_z} + \mu {H_y}{H_z}}\\ {\varepsilon {E_x}{E_z} + \mu {H_x}{H_z}}&{\varepsilon {E_y}{E_z} + \mu {H_y}{H_z}}&{\varepsilon E_z^2 + \mu H_z^2 - \frac{1}{2}({\varepsilon {{|{\vec{E}} |}^2} + \mu {{|{\vec{H}} |}^2}} )} \end{array}} \right)$}$$

Since the average FWHM of the achromatic metalens is 797 nm, the radius of the particle was set to 350 nm. The trapping environment is air and the refractive index is 1. The incident light power is set as 100 mW for all wavelengths. Figure 5 shows the optical force of the SiO₂ particles when moving along the x-direction under both the achromatic metalens and dispersive metalens, the original position of particle is set as (0, 0, 3 µm). The Fx direction is positive when particles are in the negative semi-axis of x, while the Fx direction is negative for particles being in the positive semi-axis. Figure 5(a)-(b) show that the achromatic and dispersive metalens can both trap SiO₂ particles in x-direction under different wavelengths. In comparison, for the achromatic metalens, the optical force of the SiO₂ particles does not fluctuate significantly at different wavelengths. Then, the motion state of SiO₂ particles could remain stable. But for the dispersive metalens, the focal length cannot be well stabilized at z = 3 µm with the variation wavelength, resulting in an apparent difference in the optical force of the SiO₂ particles. Figure 5(c)-(d) show a comparison of the maximum optical force in the positive and negative x-direction of the SiO₂ particles at different wavelengths under both the achromatic and dispersive metalens, respectively. Due to the symmetrical structure of the metalens, the maximum forces on the SiO₂ particles in the positive and negative x-directions were the same. For achromatic metalens the average maximum Fx was 0.88 pN, the average optical trap stiffness [50] k_x was 26.51 pN µm⁻¹ W⁻¹, and the average trapping efficiency [55] was 0.27%. In addition, the difference between the maximum Fx was 0.26 pN. However, for the dispersive metalens case, the difference was 0.74 pN, which is significantly higher than that of the achromatic metalens. Therefore, in the x-direction, the achromatic metalens is more conducive to constant the Fx and the motion state of the trapped particles.

Fig. 5. (a-b) The optical force of the achromatic metalens and the dispersive metalens when the SiO₂ particles move in the x-direction. (c-d) Comparison of the maximum optical force in the x-direction of the achromatic metalens and the dispersive metalens when the SiO₂ particles move along the x-direction.

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Figure 6(a-b) shows the axial optical force Fz of the SiO₂ particles moving along the z-direction at different wavelengths under both the achromatic metalens and the dispersive metalens. The positive z-axis is defined as the z-coordinate value greater than 3 µm and the negative z-axis is less than 3 µm. the optical force is positive when particles are in the negative z-axis, while the optical force is negative when particles are in the positive z-axis. At different wavelengths, both the achromatic metalens and dispersive metalens can trap SiO2 particles in the z-direction. In the achromatic metalens, the equilibrium points in the z-direction are more concentrated than that within the dispersive metalens, so the difference between the trapped positions is smaller, which reduces the influence of the wavelength bandwidth on the trapped positions of the SiO₂ particles. Furthermore, for the achromatic metalens, there was a small difference in the optical force at different wavelengths, while there was an evident difference for the dispersive metalens. However, it is worth noting that for the achromatic metalens when the SiO₂ particles are in the same position, there is a difference in Fz, which is because the achromatic metalens does not eliminate dispersion. Despite this, the effect of the optimized achromatic metalens is already significantly better than that of the dispersive metalens. Figure 6(c)-(d) show the comparison of the maximum optical force in the positive and negative z-direction of the SiO₂ particles at different wavelengths, respectively, under the achromatic metalens and the dispersive metalens. For the achromatic metalens, the average maximum positive Fz was 0.72 pN, the average maximum negative Fz was 0.52 pN, the average k_z were 10.74 pN µm⁻¹ W⁻¹, and the trapping efficiency was 0.22%. Furthermore, the maximum positive and negative Fz is more concentrated at different wavelengths, and the difference between the maximum positive and negative Fz were 0.17 pN and 0.06 pN, respectively. While the difference in the maximum positive and negative Fz of the dispersive metalens was 0.33 pN and 0.3 pN, which is significantly higher than that of the achromatic metalens. Figure 6(e) shows a comparison of the equilibrium points of the achromatic metalens and the dispersive metalens. The equilibrium points of the achromatic metalens are more concentrated, and the difference was 0.72 µm. However, that of the dispersive metalens was 1.05 µm, which is larger than that of the achromatic metalens. Therefore, in the z-direction, the achromatic metalens is more beneficial to maintaining the constant of Fz and the stability of trapped position.

Fig. 6. (a-b) The optical force of achromatic metalens and the dispersive metalens when the SiO2 particles move along the z-direction. Comparison of the maximum force in the positive (c) and the negative z-directions (d) and the equilibrium point position (e) of SiO2 particles under the achromatic metalens and the dispersive metalens. (f-e) Trap stiffness at different wavelengths. (h) Trapping efficiency at different wavelengths.

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Table 2 shows the comparison of the optical trap stiffness and trapping dimension of the achromatic metalens with literature. The stiffness of the optical trap in this work is larger than that of the literature, and the trapping dimension is 3D.

Table 2. Trapping dimension and optical trap stiffness k of achromatic metalens compared to literature

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For the adhesion between the particle and the metalens surface, which may hinder the trapping of particles. Therefore, we calculate the van der Waals (F_vdw) force between the particle and the surface of the metalens. For sphere-ideal plane contact the classical van der Waals force formula is as follows [56]:

(10)$${F_{vdw}} = \frac{{HR}}{{6{a^2}}}$$

Here, H is the Hamaker constant, R is the radius of the sphere, and a is the distance of the sphere from the metalens plane. In air medium, Hamaker constant H = 10.38e⁻²⁰J, so for the achromatic metalens, ${F_{vdw}}$ is 6.7278e⁻⁷pN, which is much smaller than the optical force. Therefore, the viscous drag of the particle on the surface can be ignored.

To evaluate the stability of trapping SiO₂ particles at the equilibrium point, we used integrated optical tweezers to calculate the potential depth of SiO₂ particles at different positions using the following formula:

(11)$$U ={-} \int_\infty ^x {F(x )dx} $$

Here, U is the potential depth and x is the position of the SiO₂ particle. When the potential depth is greater than K_BT, it can be considered that the SiO₂ particles can be trapped stably, where K_B is Boltzmann's constant, and T is the temperature, assumed to be 300 K [57]. Figure 7(a)-(b) show the potential depth of the achromatic metalens and dispersive metalens on the SiO₂ particles in the x-direction, respectively. Under different wavelengths, the potential depth of the SiO₂ particles at the equilibrium point is much larger than K_BT, meaning that the SiO₂ particles can be stably trapped. The potential depth of the achromatic lens on the SiO₂ particles at the equilibrium point in the x-direction is more concentrated with a smaller fluctuation. Figure 7(c) shows the maximum potential depth of the achromatic metalens and dispersive metalens on the SiO₂ particles at different wavelengths. The average maximum potential depth of the achromatic metalens in the x-direction was 110 K_BT, while that of the dispersive metalens was 178 K_BT. There was a gradual variation in the potential depth at the equilibrium point in the x-direction of the achromatic metalens, with a difference of 32 K_BT. That of the dispersive metalens was 87 K_BT, which is significantly higher than of the achromatic metalens. Therefore, the achromatic metalens has the advantage of constant maximum potential depth at different wavelengths in the x-direction.

Fig. 7. (a-b) The potential depth of achromatic metalens and the dispersive metalens when the SiO₂ particles move along the x-direction. (c) The maximum potential depths of SiO₂ particles at different wavelengths in the achromatic metalens and dispersive metalens along the x-direction. (d-e) The potential depth of achromatic metalens and the dispersive metalens when the SiO₂ particles move along the z-direction. (f) The maximum potential depths of the SiO₂ particles at different wavelengths in the achromatic metalens and the dispersive metalens along the z-direction.

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Figure 7(d-e) show the potential depths of the achromatic metalens and the dispersive metalens on the SiO₂ particles in the z-direction, respectively. The potential depths at the equilibrium point are much larger than K_BT, which means stable trapping of the SiO₂ particles can be achieved. However, the potential depth of the achromatic metalens at the equilibrium point in the z-direction is more concentrated over different wavelengths, while that of the dispersive metalens is more discrete. Figure 7(f) shows the maximum potential depth of the achromatic metalens and dispersive metalens on the SiO₂ particles in the z-direction at different wavelengths. The average maximum potential depth of the achromatic metalens in the z-direction was 138 K_BT, and that of the dispersive metalens in the z-direction was 233 K_BT. There was a more gradual change in the potential depth at the equilibrium point in the z-direction of the achromatic metalens, with a difference of 48 K_BT, while that of the dispersive metalens was 117 K_BT, which is much higher than the achromatic metalens. Therefore, in the z-direction, the maximum potential depth of achromatic metalens at different wavelengths is more constant. Considering the x and z-direction, the achromatic metalens can trap particles in the 3D space and well maintain the motion state and the position of particles.

3. Conclusion

In this paper, we proposed an metalens tweezer with high NA for the 3D trapping of particles, and we demonstrated that the NA can reach 0.97. We used the particle swarm algorithm to optimize the phase distribution of the metalens in order to meet the phase compensation required for different wavelengths. The results show that for the achromatic metalens, the change of the focal position is more stable, and the average focusing efficiency was 44%. We then moved the SiO₂ particles with a radius of 350 nm along the x- and z-directions of the achromatic metalens, in order to calculate the optical forces of the particles. The results showed that when the wavelength has a bandwidth, the average maximum optical force of the achromatic metalens on the SiO₂ particles in the x- and z-directions was 0.88 pN and 0.72 pN, respectively, and the optical force of the SiO₂ particles was closer at different wavelengths. The difference between the maximum Fx, positive Fz, and negative Fz were 0.26 pN, 0.17 pN and 0.06 pN, respectively. Furthermore, the difference between the equilibrium point was 0.72 µm, which was more concentrated. Compared with the dispersive metalens, our proposed achromatic metalens is more conducive for the constant optical force, motion state of trapped particles, and the stability of trapping position, which increases the practicability of metalenses and provides various ways to trap particles.

Funding

National Natural Science Foundation of China (60907003, 61805278, 61875168).

Acknowledgments

The authors acknowledge the United Microelectronics Center Co., Ltd., Chongqing for software sponsorship. we also acknowledge the Chongqing Science Funds for Distinguished Young Scientists(cstc2021jcyj-jqX0027); Innovation Research 2035 Pilot Plan of Southwest University (SWU-XDPY22012); China Postdoctoral Science Foundation (2018M633704); Innovation Support Program for Overseas Students in Chongqing (cx2021008); Foundation of NUDT (JC13-02-13, ZK17-03-01); Hunan Provincial Natural Science Foundation of China (13JJ3001); Pro-gram for New Century Excellent Talents in University (NCET-12-0142).

Disclosures

The authors declare no conflict 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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Literature	[41]	[42]	[43]	[44]	This work
NA	0.106	0.2	0.04	0.6	0.97
Focusing efficiency (%)	40	15	25	18	44

Literature	[25]	[36]	This work
Trapping dimension	3D	2D	3D
k_x (pN µm⁻¹ w⁻¹)	7.00	7.53	26.51
k_z (pN µm⁻¹ w⁻¹)	null	null	10.74

Literature	[41]	[42]	[43]	[44]	This work
NA	0.106	0.2	0.04	0.6	0.97
Focusing efficiency (%)	40	15	25	18	44

Literature	[25]	[36]	This work
Trapping dimension	3D	2D	3D
k_x (pN µm⁻¹ w⁻¹)	7.00	7.53	26.51
k_z (pN µm⁻¹ w⁻¹)	null	null	10.74

Manipulation force analysis of nanoparticles with ultra-high numerical aperture metalens

Abstract

1. Introduction

2. Methods

2.1 Achromatic metalens

2.2 Optical Force Calculation

3. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Tables (2)

Equations (11)

Optics Express