Abstract

Chu et al. constructed a kind of Ince-Gaussian modes (IGM)-based vortex array laser beams consisting of p x p embedded optical vortexes from Ince-Gaussian modes, IGep,p modes [Opt. Express 16, 19934 (2008)]. Such an IGM-based vortex array laser beams maintains its vortex array profile during both propagation and focusing, and is applicable to optical tweezers. This study uses the discrete dipole approximation (DDA) method to study the properties of the IGM-based vortex array laser tweezers while it traps dielectric particles. This study calculates the resultant force exerted on the spherical dielectric particles of different sizes situated at the IGM-based vortex array laser beam waist. Numerical results show that the number of trapping spots of a structure light (i.e. IGM-based vortex laser beam), is depended on the relation between the trapped particle size and the structure light beam size. While the trapped particle is small comparing to the beam size of the IGM-based vortex array laser beams, the IGM-based vortex array laser beams tweezers are suitable for multiple traps. Conversely, the tweezers is suitable for single traps. The results of this study is useful to the future development of the vortex array laser tweezers applications.

© 2013 Optical Society of America

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2012 (1)

B. Ma, B. Yao, F. Peng, S. Yan, M. Lei, and R. Rupp, “Optical sorting of particles by dual-channel line optical tweezers,” J. Opt.14(10), 105702 (2012).
[CrossRef]

2010 (2)

K. Dholakia and P. Zemánek, “Colloquium: Gripped by light: Optical binding,” Rev. Mod. Phys.82(2), 1767–1791 (2010).
[CrossRef]

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys.108(7), 073110 (2010).
[CrossRef]

2009 (1)

2008 (1)

2005 (2)

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol.23(1), 83–87 (2005).
[CrossRef] [PubMed]

R.-J. Yang, C.-C. Chang, S.-B. Huang, and G.-B. Lee, “A new focusing model and switching approach for electrokinetic flow inside microchannels,” J. Micromech. Microeng.15(11), 2141–2148 (2005).
[CrossRef]

2004 (2)

2003 (1)

2001 (4)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292(5518), 912–914 (2001).
[CrossRef] [PubMed]

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phys. Rev. A63(6), 063401 (2001).
[CrossRef]

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun.198(1-3), 21–27 (2001).
[CrossRef]

A. G. Hoekstra, M. Frijlink, L. B. Waters, and P. M. Sloot, “Radiation forces in the discrete-dipole approximation,” J. Opt. Soc. Am. A18(8), 1944–1953 (2001).
[CrossRef] [PubMed]

1999 (1)

1997 (1)

1996 (2)

K. T. Gahagan and G. A. Swartzlander., “Optical vortex trapping of particles,” Opt. Lett.21(11), 827–829 (1996).
[CrossRef] [PubMed]

B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains. I. superthermal spin-up,” Astrophys. J.470, 551–565 (1996).
[CrossRef]

1994 (1)

1993 (1)

B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J.405, 685–697 (1993).
[CrossRef]

1991 (1)

1988 (1)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J.333, 848–872 (1988).
[CrossRef]

1987 (1)

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science235(4795), 1517–1520 (1987).
[CrossRef] [PubMed]

1980 (1)

1971 (1)

A. Ashkin and J. M. Dziedzic, “Optical levitation by radia-tion pressure,” Appl. Phys. Lett.19(8), 283–285 (1971).
[CrossRef]

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radia-tion pressure,” Phys. Rev. Lett.24(4), 156–159 (1970).
[CrossRef]

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Ashkin, A.

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science235(4795), 1517–1520 (1987).
[CrossRef] [PubMed]

A. Ashkin and J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt.19(5), 660–668 (1980).
[CrossRef] [PubMed]

A. Ashkin and J. M. Dziedzic, “Optical levitation by radia-tion pressure,” Appl. Phys. Lett.19(8), 283–285 (1971).
[CrossRef]

A. Ashkin, “Acceleration and trapping of particles by radia-tion pressure,” Phys. Rev. Lett.24(4), 156–159 (1970).
[CrossRef]

Bandres, M. A.

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Butler, W. F.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol.23(1), 83–87 (2005).
[CrossRef] [PubMed]

Chang, C.-C.

R.-J. Yang, C.-C. Chang, S.-B. Huang, and G.-B. Lee, “A new focusing model and switching approach for electrokinetic flow inside microchannels,” J. Micromech. Microeng.15(11), 2141–2148 (2005).
[CrossRef]

Chu, S.-C.

Dees, B.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol.23(1), 83–87 (2005).
[CrossRef] [PubMed]

Dholakia, K.

K. Dholakia and P. Zemánek, “Colloquium: Gripped by light: Optical binding,” Rev. Mod. Phys.82(2), 1767–1791 (2010).
[CrossRef]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Draine, B. T.

B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains. I. superthermal spin-up,” Astrophys. J.470, 551–565 (1996).
[CrossRef]

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A11(4), 1491–1499 (1994).
[CrossRef]

B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J.405, 685–697 (1993).
[CrossRef]

J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett.16(15), 1198–1200 (1991).
[CrossRef] [PubMed]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J.333, 848–872 (1988).
[CrossRef]

Dubik, B.

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun.198(1-3), 21–27 (2001).
[CrossRef]

Dziedzic, J. M.

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science235(4795), 1517–1520 (1987).
[CrossRef] [PubMed]

A. Ashkin and J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt.19(5), 660–668 (1980).
[CrossRef] [PubMed]

A. Ashkin and J. M. Dziedzic, “Optical levitation by radia-tion pressure,” Appl. Phys. Lett.19(8), 283–285 (1971).
[CrossRef]

Flatau, P. J.

Forster, A. H.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol.23(1), 83–87 (2005).
[CrossRef] [PubMed]

Frijlink, M.

Gahagan, K. T.

Goodman, J.

B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J.405, 685–697 (1993).
[CrossRef]

Goodman, J. J.

Gutiérrez-Vega, J. C.

Hagen, N.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol.23(1), 83–87 (2005).
[CrossRef] [PubMed]

Hill Iii, W. T.

Hirleman, E. D.

Hoekstra, A. G.

Huang, L.

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys.108(7), 073110 (2010).
[CrossRef]

Huang, S.-B.

R.-J. Yang, C.-C. Chang, S.-B. Huang, and G.-B. Lee, “A new focusing model and switching approach for electrokinetic flow inside microchannels,” J. Micromech. Microeng.15(11), 2141–2148 (2005).
[CrossRef]

Jhe, W.

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phys. Rev. A63(6), 063401 (2001).
[CrossRef]

Kariv, I.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol.23(1), 83–87 (2005).
[CrossRef] [PubMed]

Kim, K.

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phys. Rev. A63(6), 063401 (2001).
[CrossRef]

Kwon, N.

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phys. Rev. A63(6), 063401 (2001).
[CrossRef]

Lee, G.-B.

R.-J. Yang, C.-C. Chang, S.-B. Huang, and G.-B. Lee, “A new focusing model and switching approach for electrokinetic flow inside microchannels,” J. Micromech. Microeng.15(11), 2141–2148 (2005).
[CrossRef]

Lei, M.

B. Ma, B. Yao, F. Peng, S. Yan, M. Lei, and R. Rupp, “Optical sorting of particles by dual-channel line optical tweezers,” J. Opt.14(10), 105702 (2012).
[CrossRef]

Li, Z.-Y.

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys.108(7), 073110 (2010).
[CrossRef]

Ling, L.

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys.108(7), 073110 (2010).
[CrossRef]

Liu, Y.

Ma, B.

B. Ma, B. Yao, F. Peng, S. Yan, M. Lei, and R. Rupp, “Optical sorting of particles by dual-channel line optical tweezers,” J. Opt.14(10), 105702 (2012).
[CrossRef]

MacDonald, M. P.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Marchand, P. J.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol.23(1), 83–87 (2005).
[CrossRef] [PubMed]

Masajada, J.

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun.198(1-3), 21–27 (2001).
[CrossRef]

Mercer, E. M.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol.23(1), 83–87 (2005).
[CrossRef] [PubMed]

Milam, D.

Nebeker, B. M.

Otsuka, K.

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Peng, F.

B. Ma, B. Yao, F. Peng, S. Yan, M. Lei, and R. Rupp, “Optical sorting of particles by dual-channel line optical tweezers,” J. Opt.14(10), 105702 (2012).
[CrossRef]

Raymond, D. E.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol.23(1), 83–87 (2005).
[CrossRef] [PubMed]

Rupp, R.

B. Ma, B. Yao, F. Peng, S. Yan, M. Lei, and R. Rupp, “Optical sorting of particles by dual-channel line optical tweezers,” J. Opt.14(10), 105702 (2012).
[CrossRef]

Schmehl, R.

Sibbett, W.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Sloot, P. M.

Song, Y.

Swartzlander, G. A.

Tu, E.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol.23(1), 83–87 (2005).
[CrossRef] [PubMed]

Wang, M. M.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol.23(1), 83–87 (2005).
[CrossRef] [PubMed]

Waters, L. B.

Weingartner, J. C.

B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains. I. superthermal spin-up,” Astrophys. J.470, 551–565 (1996).
[CrossRef]

Xu, X.

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phys. Rev. A63(6), 063401 (2001).
[CrossRef]

Yan, S.

B. Ma, B. Yao, F. Peng, S. Yan, M. Lei, and R. Rupp, “Optical sorting of particles by dual-channel line optical tweezers,” J. Opt.14(10), 105702 (2012).
[CrossRef]

Yang, C.-S.

Yang, J. M.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol.23(1), 83–87 (2005).
[CrossRef] [PubMed]

Yang, R.-J.

R.-J. Yang, C.-C. Chang, S.-B. Huang, and G.-B. Lee, “A new focusing model and switching approach for electrokinetic flow inside microchannels,” J. Micromech. Microeng.15(11), 2141–2148 (2005).
[CrossRef]

Yao, B.

B. Ma, B. Yao, F. Peng, S. Yan, M. Lei, and R. Rupp, “Optical sorting of particles by dual-channel line optical tweezers,” J. Opt.14(10), 105702 (2012).
[CrossRef]

Yu, M.

Yuan, X.-C.

Zemánek, P.

K. Dholakia and P. Zemánek, “Colloquium: Gripped by light: Optical binding,” Rev. Mod. Phys.82(2), 1767–1791 (2010).
[CrossRef]

Zhang, D. W.

Zhang, H. C.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol.23(1), 83–87 (2005).
[CrossRef] [PubMed]

Zhou, F.

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys.108(7), 073110 (2010).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Ashkin and J. M. Dziedzic, “Optical levitation by radia-tion pressure,” Appl. Phys. Lett.19(8), 283–285 (1971).
[CrossRef]

Astrophys. J. (3)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J.333, 848–872 (1988).
[CrossRef]

B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J.405, 685–697 (1993).
[CrossRef]

B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains. I. superthermal spin-up,” Astrophys. J.470, 551–565 (1996).
[CrossRef]

J. Appl. Phys. (1)

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys.108(7), 073110 (2010).
[CrossRef]

J. Micromech. Microeng. (1)

R.-J. Yang, C.-C. Chang, S.-B. Huang, and G.-B. Lee, “A new focusing model and switching approach for electrokinetic flow inside microchannels,” J. Micromech. Microeng.15(11), 2141–2148 (2005).
[CrossRef]

J. Opt. (1)

B. Ma, B. Yao, F. Peng, S. Yan, M. Lei, and R. Rupp, “Optical sorting of particles by dual-channel line optical tweezers,” J. Opt.14(10), 105702 (2012).
[CrossRef]

J. Opt. Soc. Am. A (4)

Nat. Biotechnol. (1)

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol.23(1), 83–87 (2005).
[CrossRef] [PubMed]

Opt. Commun. (1)

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun.198(1-3), 21–27 (2001).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Amplitude distribution of even Ince-Gaussian beams, IGep,p, and its corresponding IGM-based vortex array laser beams with mode order: (a) p = 2, (b) p = 4 and (c) p = 6.

Fig. 2
Fig. 2

(a) Extinction efficiency Qext at different size parameter x( = 2πa/λ). (b) Error in Qext at different size parameter x = 2πa/λ. (c) Scattering efficiency times the asymmetry parameter, gQsca, at different size parameter x = 2πa/λ. (d) Error in gQsca at different size parameter.

Fig. 3
Fig. 3

Simulation results of a p = 2 IGM-based vortex array laser tweezers with different-size particles: (b) a = 0.1 µm, (c) a = 0.3 µm, (d) a = 0.5 µm, (e) a = 0.7 µm, and (f) a = 1.0 µm. Figure 3(a) plots the intensity distribution of a vortex array laser beam constructed by p = 2 even Ince-Gaussian beams, IGe2,2 mode. The color backgrounds in Figs. 3(b)3(f) plot the distributions of the absolute value of the resultant force on the trapped particle to its x-y position. The dark arrows show the vector of resultant force acting on the particle while the particle is situated at the arrow tail. In simulations, the beam waist of the p = 2 Ince-Gaussian beams is 1.0µm.

Fig. 4
Fig. 4

Simulation results of a p = 4 IGM-based vortex array laser tweezers with different-size particles: (b) a = 0.1 µm, (c) a = 0.3 µm, (d) a = 0.5 µm, (e) a = 1.0 µm, and (f) a = 1.5 µm. Figure 4(a) plots the intensity distribution of a vortex array laser beam constructed by p = 4 even Ince-Gaussian beams, IGe4,4 mode. The color backgrounds in Figs. 4(b)4(f) plot the distributions of the absolute value of the resultant force on the trapped particle to its x-y position. The dark arrows show the vector of resultant force acting on the particle while the particle is situated at the arrow tail. In simulations, the beam waist of the p = 4 Ince-Gaussian beams is 1.0µm.

Fig. 5
Fig. 5

The ratio of peak scattering force to peak incident force of p = 2 and p = 4 vortex array laser tweezers at different particle radius. Here, the beam waist of the composite Ince-Gaussian beams is 1.0µm.

Fig. 6
Fig. 6

Simulated resultant force on a trapped particle of two situations. Red lines show the results of the first situation, i.e., the resultant force on a single particle of radius a1 under the vortex array laser beam. The blue lines show the results of the other situation, i.e., the resultant force on a small particle of radius a1 under the vortex array laser beam, while an additional trapped particle of radius a2 exists at the beam center. Three examples are calculated: (a) a1 = 0.1 μm, a2 = 1.5 μm, (b) a1 = 0.1 μm, a2 = 1.0 μm, and (c) a1 = 0.3 μm, a2 = 1.0 μm.

Fig. 7
Fig. 7

Simulated resultant force distribution on the trapped particle of the p = 4 IGM-based vortex array laser tweezers with different IG beam waist w0 and different trapped particle radius a: (a) w0 = 1.0μm, a = 0.1 µm, (b) w0 = 10μm, a = 1.0 µm, (c) w0 = 1.0μm, a = 0.3 µm and (d) w0 = 3.0μm, a = 0.9 µm. In the situations of Figs. 7(a) and 7(b), the ratio a/w0 is 0.1. In the situations of Figs. 7(c) and 7(d), the ratio a/w0 is 0.3. The color backgrounds plot the distributions of the absolute value of the resultant force on the trapped particle to its x-y position. The dark arrows show the vector of resultant force acting on the particle while the particle is situated at the arrow tail. In both simulations, the laser beam wavelength λ is 1.064μm.

Equations (10)

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( U VL ) p,p =I G p,p e +i× [ I G p,p e ] T ,
I G p,m e ( r,ε )=C[ w 0 / w( z ) ] C p m ( iξ,ε ) C p m ( η,ε )exp[ r 2 / w 2 ( z ) ] ×expi[ kz+ k r 2 / 2R( z ) ( p+1 ) ψ z ( z ) ],
P i = α i E tot,i ,
E tot,i = E inc,i j=1,ji N G ij P j ,
j N A ij P ij = E inc,i ( i=1,,N ),
F i = 1 2 Re[ ( P i * i ) E i +ik P i * × B i ],
F i = F inc,i + F sca,i ,
F inc = 1 8π C ext | E inc | 2 ,
F sca = 1 8π g C sca | E inc | 2 ,
Q pr = Q ext g Q sca ,

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