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Abstract
We rigorously calculate the conservative gradient force (GF) and the non-conservative scattering force (SF) associated with the optical tweezers (the single beam optical trap). A wide range of parameters are considered, with particle size ranging from the Rayleigh to Mie regime (radius ∼3 µm), dielectric constant ranging from metallic (large and negative) to high dielectrics (large and positive), numerical aperture (NA) ranging from 0.5 to 1.33, and different polarizations. The trap depth associated with GF can reach 123 and 168 kBT per mW for a 0.5 µm-radius polystyrene particle illuminated by a 1064 nm Gaussian beam with NA = 0.9 and 1.3, respectively. This indicates that unless at a low beam power or with a small NA, the Brownian fluctuations do not play a role in the stability. The transverse GF orthogonal to beam propagation always dominates over the transverse SF. While the longitudinal SF can be larger than the longitudinal GF when the scattering is strong, the NA is small, or when absorption is present, optical trapping under these conditions is difficult. Generally speaking, absorption reduces GF and enhances SF, while increasing a dielectric constant enhances GF slightly but boosts SF significantly owing to stronger scattering. These results verify previous experimental observations and explain why optical tweezers are so robust across such a wide range of conditions. Our quantitative calculations will also provide a guide to future studies.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Invented in 1986 [1], optical tweezers has proven itself to be an invaluable tool for manipulating small particles [2–7], owing to its simplicity and versatility [8–24]. According to the Helmholtz theorem [25], it is possible and useful to divide the optical force acting on a single particle into two mathematically and physically distinct components:
The concept of GF and SF are widely employed in the literature to interpret experimental and theoretical results qualitatively. Yet, for a long time, their true profiles were unknown except for the limiting cases of particle small [1,45–50] or large [51,52] compare to the light wavelength. Recently, some examples of the GF and SF for Mie sized particles [53] were calculated [44,54–57], however, no systematic study on optical tweezers is available. Here, by adopting the fast Fourier transform approach presented in Ref. [44], we explicitly reveal the long missing profile of GF and SF for optical tweezers under a wide range of operational parameters.
2. Methodology
The total (time-averaged) optical force $\textbf{F}(\textbf{x} )$ can be obtained by integrating the time-averaged Maxwell stress tensor $\overset\leftrightarrow{\textrm{T}}$ [25] over a surface S that encloses the particle:
To generate a strong intensity gradient for trapping, it is probably most convenient to use a Gaussian beam focused by a high numerical aperture (NA) objective lens. Throughout this paper, if not otherwise stated, a z-propagating and x-polarized Gaussian beam with a wavelength of 1064 nm is adopted, and the 1 mW beam is focused by a NA = 1.3 water immersion lens into water (${n_\textrm{w}} = 1.33$). The trapping is assumed to be conducted at room temperature T = 300 K, and the thermal energy is ${k_\textrm{B}}T$=4.412${\times} {10^{ - 21}}$J, where ${k_\textrm{B}}$ is the Boltzmann constant.
3. GF and SF for optical tweezers
The GF and SF for a 0.5 µm-radius polystyrene (dielectric constant of 1.572) sphere are plotted in Fig. 1 for NA = 1.3 and Fig. 2 for NA = 0.9. The potential energy for GF, defined by
Fig. 1. GF and SF acting on a 0.5 µm-radius polystyrene sphere (${\varepsilon _r} = {1.57^2}$) immersed in water. (a)-(c) are the contour plots of potential energy (U) in different planes. In plane SF is expressed as black arrows in (d)-(f), where the longest arrow has a magnitude indicated at the bottom of each figure, while SFz is shown with color scale. GFz (black line) and SFz (white line) along the beam axis are plotted in (b). The power of the incident beam is 1 mW, while the numerical aperture (NA) is 1.3.
Fig. 2. GF and SF acting on a 0.5 µm-radius polystyrene bead (${\varepsilon _r} = {1.57^2}$) immersed in water. (a)-(c) are the contour plots of potential energy (U) in different planes. In plane SF is expressed as black arrows in (d)-(f), where the longest arrow has a magnitude indicated at the bottom of each figure, while the SFz is shown with color scale. GFz (black line) and SFz (white line) along the beam axis are plotted in (b). The power of the incident beam is 1 mW, while the numerical aperture (NA) is 0.9.
Near the focus, the constant potential energy surfaces are approximately ellipsoids. The gradients of U along the x and y axes are approximately the same, while the slope along the z axis is a factor of two smaller. SF is plotted in panel (d)-(f) of Fig. 1 and Fig. 2. The arrows represent in-plane SF and the colored background contour plots represent the out of plane component. SF is strongest near the beam center and is pointing along the beam’s propagation direction.
GFz (black) and SFz (white) along the beam axis are plotted in Fig. 1(b) and Fig. 2(b). For NA = 1.3, GFz is roughly twice the SFz, while for NA = 0.9, they are comparable. The equilibrium position for optical trapping would be along the beam axis and satisfied GFz+SFz = 0. Accordingly, the equilibrium position (if any) is not exactly located at the potential energy minimum of GF at the focus due to the push from SFz. As shown in panel (d)-(e) of Fig. 1 and Fig. 2, SF has a tendency to twirl the particle, not just along the z-axis as reported in Ref. [26], but also on the xy-plane. One could observe in Fig. 1(b) that the SFz (white curve) has a local minimum at the beam center. Such phenomenon is due to high transmission of light when the particle is at the origin, as discussed in Ref. [58].
GF and SF for a slightly absorptive particle are plotted in Fig. 3, where the dielectric constant is 1.572+0.1i and everything else are the same as Fig. 1. GF is decreased by ∼20-25%, while SF is increased by several times. That explains why absorptive particles are difficult to trap. SF and GF for the high dielectric Mie particle, metallic dipolar particle, and metallic Mie particles are plotted in Fig. 4, Fig. 5, and Fig. 6, respectively. As shown in Fig. 4(b), for the high dielectric particle with $r = 0.50\; \mathrm{\mu}\textrm{m}$ and ${\varepsilon _r} = 9$, SFz (white line) is stronger than GFz (black line) along the beam axis, making trapping impossible. Compare to polystyrene, GF of high dielectrics is enhanced by more than 50% while SF is enhanced by several times. Also, as clearly seen in Fig. 4(a), the constant potential energy surfaces deviate from ellipsoids. From Fig. 4(f), the asymmetry between x and y axes can be large.
Fig. 3. GF and SF acting on a 0.5 µm-radius, slightly absorptive sphere (${\varepsilon _r} = {1.57^2} + 0.1i$) immersed in water. (a)-(c) are the contour plots of potential energy (U) in different planes. In plane SF is expressed as black arrows in (d)-(f), where the longest arrow has a magnitude indicated at the bottom of each figure, while SFz is shown with color scale. GFz (black line) and SFz (white line) along the beam axis are plotted in (b). The power of the incident beam is 1 mW, while the numerical aperture (NA) is 1.3.
Fig. 4. Optical forces acting on a 0.5 µm-radius high-dielectric (${\varepsilon _r} = 9$) sphere immersed in water. (a)-(c) are the contour plots of potential energy (U) in different planes. In plane SF is expressed as black arrows in (d)-(f), where the longest arrow has a magnitude indicated at the bottom of each figure, while SFz is shown with color scale. GFz (black line) and SFz (white line) along the beam axis are plotted in (b). The power of the incident beam is 1 mW, while the numerical aperture (NA) is 1.3.
Fig. 5. Optical forces acting on a 0.1 µm-radius gold (${\varepsilon _r} ={-} 48.45 + 3.6i$) sphere immersed in water. (a)-(c) are the contour plots of potential energy (U) in different planes. In plane SF is expressed as black arrows in (d)-(f), where the longest arrow has a magnitude indicated at the bottom of each figure, while the SFz is shown with color scale. GFz (black line) and SFz (white line) along the beam axis are plotted in (b). The power of the incident beam is 1 mW, while the numerical aperture (NA) is 1.3.
Fig. 6. Optical forces acting on a 0.5 µm-radius gold (${\varepsilon _r} ={-} 48.45 + 3.6i$) sphere immersed in water. (a)-(c) are the contour plots of potential energy (U) in different planes. In plane SF is expressed as black arrows in (d)-(f), where the longest arrow has a magnitude indicated at the bottom of each figure, while the SFz is shown with color scale. GFz (black line) and SFz (white line) along the beam axis are plotted in (b). The power of the incident beam is 1 mW, while the numerical aperture (NA) is 1.3.
For a dipolar metallic particle ($r = 0.10\; \mathrm{\mu}\textrm{m}$), the GF confines the particle to the beam center (black curve in Fig. 5(b)), allowing stable transverse trapping. However, for metallic Mie particle ($r = 0.50\; \mathrm{\mu}\textrm{m}$), the GF expels the particle from the beam center (black curve in Fig. 6(b)), making trapping impossible.
We remark that the GF has odd symmetry along x, y, and z, while the SF is odd symmetric along x and y but is even symmetric along z (data not shown). Such “hidden symmetries”, namely SF and GF have more symmetries than the original system, are discussed in Ref. [56].
4. Strength of GF and SF
We shall calculate and study the global maximum of GF and SF under a wide range of parameters and situations. Figure 7(a) plots the global maximum of the cartesian components of GF and SF for a polystyrene sphere versus particle radius r, which ranges from 0.01 µm (dipolar particle) to 3 µm (Mie particle). Both GF and SF are increasing monotonically. However, while GF saturates at r ∼ 2 µm, SF keeps increasing for the entire range, although its pace has slowed down for r > 0.20 µm. For r < 0.10 µm, since GF${\propto} {r^3}$ and SF${\propto} {r^6}$ for Rayleigh particles, GF>>SF [1]. In short, the GF is always dominating over the SF, except for particles larger than ∼0.5µm in radius, where the longitudinal SF is larger than the longitudinal GF owing to the strong scattering.
Fig. 7. (a) Global maximum of the GF and SF versus the particle radius. (b) Asymmetry of GF in the x and y directions. The green arrow marks the particle radius corresponding to Fig. 1.
When the x-polarized Gaussian beam is strongly focused, its focus is slightly elongated along the x-direction (see Fig. 1(f)), resulting in an asymmetry between GFx and GFy (also SFx and SFy), especially for small particles, as illustrated in Fig. 7(b).
NA represents the focusing power of the lens. Altering NA will change the trap’s energy depth, spatial extent, strength, etc., as evident in Fig. 1 and Fig. 2. The global maximum of GF and SF are plotted in Fig. 8 for NA ranging from 0.5 to 1.33, and $r = 0.01,\; 0.5,\; \textrm{and}\; 1.0\; \mathrm{\mu}\textrm{m}$. Both GF and SF increase approximately linearly with NA in the log-log scale, except for the transverse SF with $r = 0.01\; \mathrm{\mu}\textrm{m}$, which is very small. The GFs grow faster than the SFs: increasing NA enhances both intensity and its gradient, while both factors favor GF, only the former one favors SF. In short, the transverse force is dominated by GF, while the longitudinal force is dominated by SF for low NAs and large particles. Moreover, for high NA, the potential energy on the xy-plane is also anisotropic, because the difference between GFx and GFy (and also between SFx and SFy) increases as NA increases.
Fig. 8. Maximum of GF and SF in different directions, where (a) $r = 0.01\; \mathrm{\mu}\textrm{m}$, (b) $r = 0.50\; \mathrm{\mu}\textrm{m}$, and (c) $r = 1.00\; \mathrm{\mu}\textrm{m}$. The numerical aperture (NA) varies from 0.5 to 1.33. The green arrow marks the NA corresponding to Fig. 2.
Figure 9 plots the maximum GFs and SFs for ${\varepsilon _r} \in [{2,9} ]$ and ${\varepsilon _r} \to \infty $. The forces are not increasing monotonically with ${\varepsilon _r}$. For $r = 0.01\; \mathrm{\mu}\textrm{m}$ (dipole), GF exceeds SF by ∼3 orders of magnitude. For larger particles ($r = 0.5\; \textrm{and}\; 1.0\; \mathrm{\mu}\textrm{m}$), the transverse GF is still much stronger than the transverse SF. For large dielectric constant, the longitudinal SF is significantly larger than the longitudinal GF, whereas the transverse GF is larger than the transverse SF.
Fig. 9. Global maximum of GF and SF along different directions versus ${\varepsilon _r}$. (a) GF for $r = 0.01\; \mathrm{\mu}\textrm{m}$, (b) SF for $r = 0.01\; \mathrm{\mu}\textrm{m}$, (c) GF and SF for $r = 0.50\; \mathrm{\mu}\textrm{m}$, and (d) GF and SF for $r = 1.00\; \mathrm{\mu}\textrm{m}$. Green arrow marks the ${\varepsilon _r}$ corresponding to Fig. 4. The asymptotic values of the forces as ${\varepsilon _r}$ approaches to infinity are also shown in the insets of (a), (b), (c) and (d).
5. GF and SF for a circularly polarized focused beam
We also considered the profiles of GF and SF for a circularly polarized focused beam, which is shown in Fig. 10. Except for the polarization, all other parameters are all the same as Fig. 1. Through comparing (a)-(c) in Fig. 1 (linear polarization) and Fig. 10 (circular polarization), we find that the profiles of potential energy and the amplitude of GFz and SFz along beam axis are almost the same. The most obvious difference between Fig. 1 and Fig. 10 is the “force vortex” on the transverse xy-plane, as shown in Fig. 10(f) with black arrows and Fig. 10(d-e) with contour plots. This “force vortex” is due to the conservation of spin to orbital angular momentum after the beam being focused. The second difference is the axial-symmetric SFz(x,y) in xy-plane for circularly polarized beam (Fig. 10(f)), while SFz(x,y) is anisotropic for linearly polarized beam (Fig. 1(f)).
Fig. 10. GF and SF acting on a 0.5 µm-radius polystyrene sphere (${\varepsilon _r} = {1.57^2}$) immersed in water. (a)-(c) are the contour plots of potential energy (U) in different planes. In plane SF is expressed as black arrows in (d)-(f), where the longest arrow has a magnitude indicated at the bottom of each figure, while SFz is shown with color scale. GFz (black line) and SFz (white line) along the beam axis are plotted in (b). The incident beam is circularly polarized while the beam power is 1 mW. The numerical aperture (NA) is 1.3.
6. Conclusions
We present the profiles of GF and SF for a wide range of trapping conditions and parameters. In particular, we investigate the strength of GF and SF versus particle radius, NA, and dielectric constant. We conclude that (1) unless the beam power or the NA is unreasonably low, Brownian fluctuations do not play a role for the stability of Mie sized particles, as the trapping potential can easily reach 100 ${k_\textrm{B}}T$ or beyond for a focused beam with a time-averaged incident power of 1.0 mW, (2) the transverse GF is always greater than the transverse SF in all cases we considered, (3) the SFz is stronger than the GFz only when the NA is low so that the GF is small, or the scattering and absorption are strong so that the SFz is large. Our methodology is applicable to any isotropic particles including the chiral ones [43,57], and may bring insight to the enantioselective separation [59,60].
Funding
Research Grants Council, University Grants Committee (AoE/P-02/12, C6013-18G); National Natural Science Foundation of China (11304260, 11674204, 12074084, 12074169).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available at this time but may be obtained from the authors upon reasonable request.
References
1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef]
2. S. M. Block, “Making light work with optical tweezers,” Nature 360(6403), 493–495 (1992). [CrossRef]
3. A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U. S. A. 94(10), 4853–4860 (1997). [CrossRef]
4. P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88(12), 123601 (2002). [CrossRef]
5. A. Rohrbach, “Stiffness of optical traps: quantitative agreement between experiment and electromagnetic theory,” Phys. Rev. Lett. 95(16), 168102 (2005). [CrossRef]
6. M. L. Juan, R. Gordon, Y. Pang, F. Eftekhari, and R. Quidant, “Self-induced back-action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009). [CrossRef]
7. M. J. Padgett, J. Molloy, and D. McGloin, “Optical Tweezers: methods and applications,” CRC Press (2010).
8. K. Dholakia, M. Macdonald, and G. Spalding, “Optical tweezers: the next generation,” Phys. World 15(10), 31–35 (2002). [CrossRef]
9. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef]
10. K. Dholakia and P. Reece, “Optical micromanipulation takes hold,” Nano Today 1(1), 18–27 (2006). [CrossRef]
11. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2(1), 021875 (2008). [CrossRef]
12. K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At., Mol., Opt. Phys. 56, 261–337 (2008). [CrossRef]
13. K. Dholakia, P. Reece, and M. Gu, “Optical micromanipulation,” Chem. Soc. Rev. 37(1), 42–55 (2008). [CrossRef]
14. J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010). [CrossRef]
15. V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant optical manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010). [CrossRef]
16. V. G. Shvedov, C. Hnatovsky, A. V. Rode, and W. Krolikowski, “Robust trapping and manipulation of airborne particles with a bottle beam,” Opt. Express 19(18), 17350–17356 (2011). [CrossRef]
17. O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8(11), 807–819 (2013). [CrossRef]
18. V. Svak, O. Brzobohatý, M. Šiler, P. Jákl, J. Kaňka, P. Zemánek, and S. H. Simpson, “Transverse spin forces and non-equilibrium particle dynamics in a circularly polarized vacuum optical trap,” Nat. Commun. 9(1), 5453 (2018). [CrossRef]
19. L. P. García, J. D. Pérez, G. Volpe, A. V. Arzola, and G. Volpe, “High-performance reconstruction of microscopic force fields from Brownian trajectories,” Nat. Commun. 9(1), 4868 (2018). [CrossRef]
20. F. Nan and Z. Yan, “Silver-nanowire-based interferometric optical tweezers for enhanced optical trapping and binding of nanoparticles,” Adv. Funct. Mater. 29(7), 1808258 (2019). [CrossRef]
21. U. G. Būtaitė, G. M. Gibson, Y. L. D. Ho, M. Taverne, J. M. Taylor, and D. B. Phillips, “Indirect optical trapping using light driven micro-rotors for reconfigurable hydrodynamic manipulation,” Nat. Commun. 10(1), 1215 (2019). [CrossRef]
22. F. Kalantarifard, P. Elahi, G. Makey, O. M. Maragò, FÖ Ilday, and G. Volpe, “Intracavity optical trapping of microscopic particles in a ring-cavity fiber laser,” Nat. Commun. 10(1), 2683 (2019). [CrossRef]
23. Y. Arita, S. H. Simpson, P. Zemánek, and K. Dholakia, “Coherent oscillations of a levitated birefringent microsphere in vacuum driven by nonconservative rotation-translation coupling,” Sci. Adv. 6(23), eaaz9858 (2020). [CrossRef]
24. X. Shan, F. Wang, D. Wang, S. Wen, C. Chen, X. Di, P. Nie, J. Liao, Y. Liu, L. Ding, P. J. Reece, and D. Jin, “Optical tweezers beyond refractive index mismatch using highly doped upconversion nanoparticles,” Nat. Nanotechnol. 16(5), 531–537 (2021). [CrossRef]
25. J. D. Jackson, “Classical electrodynamics,” Wiley3rd Ed. (1999).
26. A. Ashkin and J. P. Gordon, “Stability of radiation-pressure particle traps: an optical Earnshaw theorem,” Opt. Lett. 8(10), 511–513 (1983). [CrossRef]
27. P. M. Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50(5), 702–708 (2000). [CrossRef]
28. A. Mazolli, P. A. Maia Neto, and H. M. Nussenzveig, “Theory of trapping forces in optical tweezers,” Proc. R. Soc. London, Ser. A 459(2040), 3021–3041 (2003). [CrossRef]
29. N. B. Viana, A. Mazolli, P. A. Maia Neto, H. M. Nussenzveig, M. S. Rocha, and O. N. Mesquita, “Absolute calibration of optical tweezers,” Appl. Phys. Lett. 88(13), 131110 (2006). [CrossRef]
30. R. S. Dutra, N. B. Viana, P. M. Neto, and H. M. Nussenzveig, “Polarization effects in optical tweezers,” J. Opt. A: Pure Appl. Opt. 9(8), S221–S227 (2007). [CrossRef]
31. N. B. Viana, M. S. Rocha, O. N. Mesquita, A. Mazolli, P. M. Neto, and H. M. Nussenzveig, “Towards absolute calibration of optical tweezers,” Phys. Rev. E 75(2), 021914 (2007). [CrossRef]
32. Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: The fountain of probability,” Phys. Rev. Lett. 101(12), 128301 (2008). [CrossRef]
33. D. Haefner, S. Sukhov, and A. Dogariu, “Conservative and nonconservative torques in optical binding,” Phys. Rev. Lett. 103(17), 173602 (2009). [CrossRef]
34. P. Wu, R. Huang, C. Tischer, A. Jonas, and E. L. Florin, “Direct measurement of the nonconservative force field generated by optical tweezers,” Phys. Rev. Lett. 103(10), 108101 (2009). [CrossRef]
35. S. H. Simpson and S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E 82(3), 031141 (2010). [CrossRef]
36. R. S. Dutra, N. B. Viana, P. M. Neto, and H. M. Nussenzveig, “Absolute calibration of forces in optical tweezers,” Phys. Rev. A 90(1), 013825 (2014). [CrossRef]
37. A. Irrera, A. Magazzù, P. Artoni, S. H. Simpson, S. Hanna, P. H. Jones, F. Priolo, P. G. Gucciardi, and O. M. Maragò, “Photonic torque microscopy of the nonconservative force field for optically trapped silicon nanowires,” Nano Lett. 16(7), 4181–4188 (2016). [CrossRef]
38. S. Sukhov and A. Dogariu, “Non-conservative optical forces,” Rep. Prog. Phys. 80(11), 112001 (2017). [CrossRef]
39. K. Diniz, R. S. Dutra, L. B. Pires, N. B. Viana, H. M. Nussenzveig, and P. M. Neto, “Negative optical torque on a microsphere in optical tweezers,” Opt. Express 27(5), 5905–5917 (2019). [CrossRef]
40. J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5(9), 531–534 (2011). [CrossRef]
41. X. Li, J. Chen, Z. Lin, and J. Ng, “Optical pulling at macroscopic distances,” Sci. Adv. 5(3), eaau7814 (2019). [CrossRef]
42. M. Mangeat, Y. Amarouchene, Y. Louyer, T. Guérin, and D. S. Dean, “Role of nonconservative scattering forces and damping on Brownian particles in optical traps,” Phys. Rev. E 99(5), 052107 (2019). [CrossRef]
43. H. Zheng, X. Li, J. Ng, H. Chen, and Z. Lin, “Tailoring the gradient and scattering forces for longitudinal sorting of generic-size chiral particles,” Opt. Lett. 45(16), 4515–4518 (2020). [CrossRef]
44. J. Du, C.-H. Yuen, X. Li, K. Ding, G. Du, Z. Lin, C. T. Chan, and J. Ng, “Tailoring optical gradient force and optical scattering and absorption force,” Sci. Rep. 7(1), 1–7 (2017). [CrossRef]
45. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996). [CrossRef]
46. A. Rohrbach and E. H. Stelzer, “Optical trapping of dielectric particles in arbitrary fields,” J. Opt. Soc. Am. A 18(4), 839–853 (2001). [CrossRef]
47. J. R. Arias-González and M. Nieto-Vesperinas, “Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A 20(7), 1201–1209 (2003). [CrossRef]
48. S. Albaladejo, M. I. Marqués, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102(11), 113602 (2009). [CrossRef]
49. D. B. Ruffner and D. G. Grier, “Comment on “Scattering forces from the curl of the spin angular momentum of a light field”,” Phys. Rev. Lett. 111(5), 059301 (2013). [CrossRef]
50. K. Ding, J. Ng, L. Zhou, and C. T. Chan, “Realization of optical pulling forces using chirality,” Phys. Rev. A 89(6), 063825 (2014). [CrossRef]
51. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992). [CrossRef]
52. H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32(13), 1839–1841 (2007). [CrossRef]
53. Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34(21), 4573–4588 (1995). [CrossRef]
54. N. Wang, X. Li, J. Chen, Z. Lin, and J. Ng, “Gradient and scattering forces of anti-reflection-coated spheres in an aplanatic beam,” Sci. Rep. 8(1), 17423 (2018). [CrossRef]
55. H. Zheng, X. Yu, W. Lu, J. Ng, and Z. Lin, “GCforce: decomposition of optical force into gradient and scattering parts,” Comput. Phys. Commun. 237, 188–198 (2019). [CrossRef]
56. Y. Jiang, H. Lin, X. Li, J. Chen, J. Du, and J. Ng, “Hidden Symmetry and Invariance in Optical Forces,” ACS Photonics 6(11), 2749–2756 (2019). [CrossRef]
57. H. Zheng, X. Li, Y. Jiang, J. Ng, Z. Lin, and H. Chen, “General formulations for computing the optical gradient and scattering forces on a spherical chiral particle immersed in generic monochromatic optical fields,” Phys. Rev. A 101(5), 053830 (2020). [CrossRef]
58. D. Huang, P. Wan, L. Zhou, H. Guo, R. Zhao, J. Chen, J. Du, and J. Ng, “The dominant role of “quiet zones” in optical trapping of Mie particles,” (In Press).
59. Y. Shi, S. Xiong, Y. Zhang, L. K. Chin, Y.-Y. Chen, J. B. Zhang, T. H. Zhang, W. Ser, A. Larrson, S. H. Lim, J. H. Wu, T. N. Chen, Z. C. Yang, Y. L. Hao, B. Liedberg, P. H. Yap, K. Wang, D. P. Tsai, C.-W. Qiu, and A. Q. Liu, “Sculpting nanoparticle dynamics for single-bacteria-level screening and direct binding-efficiency measurement,” Nat. Commun. 9(1), 1–11 (2018). [CrossRef]
60. Y. Shi, “Chirality-assisted lateral momentum transfer for bidirectional enantioselective separation,” Light: Sci. Appl. 9(1), 1–12 (2020). [CrossRef]
