Abstract

We investigate optical torques over absorbent negative refractive index spherical scatterers under the influence of linear and circularly polarized TEM00 focused Gaussian beams, in the framework of the generalized Lorenz-Mie theory with the integral localized approximation. The fundamental differences between optical torques due to spin angular momentum transfer in positive and negative refractive index optical trapping are outlined, revealing the effect of the Mie scattering coefficients in one of the most fundamental properties in optical trapping systems.

© 2011 OSA

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  28. K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994).
    [CrossRef]
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    [CrossRef]
  36. J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. off-axis beams,” J. Opt. Soc. Am. A 11(9), 2516–2525 (1994).
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    [CrossRef]

2010 (2)

2009 (2)

A. E. Miroshnichenko, “Non-Rayleigh limit of the Lorenz-Mie solution and suppression of scattering by spheres of negative refractive index,” Phys. Rev. A 80(1), 013808 (2009).
[CrossRef]

L. A. Ambrosio and H. E. Hernández-Figueroa, “Trapping double negative particles in the ray optics regime using optical tweezers with focused beams,” Opt. Express 17(24), 21918–21924 (2009).
[CrossRef] [PubMed]

2005 (2)

N. Engheta and R. Ziolkowski, “A positive future for double-negative metamaterials,” IEEE Trans. Microw. Theory Tech. 53(4), 1535–1556 (2005).
[CrossRef]

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005).
[CrossRef] [PubMed]

2004 (2)

A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed-material plate,” Phys. Rev. Lett. 92(7), 077401 (2004).
[CrossRef] [PubMed]

D. P. O’Neal, L. R. Hirsch, N. J. Halas, J. D. Payne, and J. L. West, “Photo-thermal tumor ablation in mice using near infrared-absorbing nanoparticles,” Cancer Lett. 209(2), 171–176 (2004).
[CrossRef] [PubMed]

2003 (1)

Z. Ye, “Optical transmission and reflection of perfect lenses by left handed materials,” Phys. Rev. B 67(19), 193106 (2003).
[CrossRef]

2002 (3)

A. Grbic and G. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92(10), 5930–5935 (2002).
[CrossRef]

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66(6), 063402 (2002).
[CrossRef]

K. Volke-Sepulveda, V. Garcés-Chavez, S. Chávez-Cerda, J. Arlt, and D. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002).
[CrossRef]

2001 (2)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(5), 056625 (2001).
[CrossRef] [PubMed]

2000 (2)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

1998 (2)

H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. 155(1-3), 169–179 (1998).
[CrossRef]

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized lorenz-mie theory,” Appl. Opt. 37(19), 4218–4225 (1998).
[CrossRef] [PubMed]

1996 (1)

1994 (3)

1992 (1)

1989 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[CrossRef]

1988 (1)

B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).

1984 (1)

P. L. Marston and J. H. Crichton, “Radiation torque on a sphere caused by a circularly-polarized electromagnetic beam,” Phys. Rev. A 30(5), 2508–2516 (1984).
[CrossRef]

1982 (1)

G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).

1979 (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (1979).
[CrossRef]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik Trüber Medien, Speziell Kolloidaler Metallösungen,” Ann. Phys. 330(3), 377–445 (1908).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[CrossRef]

Alù, A.

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005).
[CrossRef] [PubMed]

Ambrosio, L. A.

Arlt, J.

K. Volke-Sepulveda, V. Garcés-Chavez, S. Chávez-Cerda, J. Arlt, and D. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002).
[CrossRef]

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[CrossRef]

Chávez-Cerda, S.

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66(6), 063402 (2002).
[CrossRef]

K. Volke-Sepulveda, V. Garcés-Chavez, S. Chávez-Cerda, J. Arlt, and D. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002).
[CrossRef]

Crichton, J. H.

P. L. Marston and J. H. Crichton, “Radiation torque on a sphere caused by a circularly-polarized electromagnetic beam,” Phys. Rev. A 30(5), 2508–2516 (1984).
[CrossRef]

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (1979).
[CrossRef]

Dholakia, D.

K. Volke-Sepulveda, V. Garcés-Chavez, S. Chávez-Cerda, J. Arlt, and D. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002).
[CrossRef]

Dholakia, K.

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66(6), 063402 (2002).
[CrossRef]

Eleftheriades, G.

A. Grbic and G. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92(10), 5930–5935 (2002).
[CrossRef]

Engheta, N.

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005).
[CrossRef] [PubMed]

N. Engheta and R. Ziolkowski, “A positive future for double-negative metamaterials,” IEEE Trans. Microw. Theory Tech. 53(4), 1535–1556 (2005).
[CrossRef]

Garcés-Chavez, V.

K. Volke-Sepulveda, V. Garcés-Chavez, S. Chávez-Cerda, J. Arlt, and D. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002).
[CrossRef]

Garcés-Chávez, V.

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66(6), 063402 (2002).
[CrossRef]

Gouesbet, G.

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized lorenz-mie theory,” Appl. Opt. 37(19), 4218–4225 (1998).
[CrossRef] [PubMed]

H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. 155(1-3), 169–179 (1998).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. off-axis beams,” J. Opt. Soc. Am. A 11(9), 2516–2525 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994).
[CrossRef]

J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. on-axis beams,” J. Opt. Soc. Am. A 11(9), 2503–2515 (1994).
[CrossRef]

B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).

G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).

Grbic, A.

A. Grbic and G. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92(10), 5930–5935 (2002).
[CrossRef]

Gréhan, G.

H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. 155(1-3), 169–179 (1998).
[CrossRef]

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized lorenz-mie theory,” Appl. Opt. 37(19), 4218–4225 (1998).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994).
[CrossRef]

B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).

G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).

Halas, N. J.

D. P. O’Neal, L. R. Hirsch, N. J. Halas, J. D. Payne, and J. L. West, “Photo-thermal tumor ablation in mice using near infrared-absorbing nanoparticles,” Cancer Lett. 209(2), 171–176 (2004).
[CrossRef] [PubMed]

Hernández-Figueroa, H. E.

Heyman, E.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(5), 056625 (2001).
[CrossRef] [PubMed]

Hirsch, L. R.

D. P. O’Neal, L. R. Hirsch, N. J. Halas, J. D. Payne, and J. L. West, “Photo-thermal tumor ablation in mice using near infrared-absorbing nanoparticles,” Cancer Lett. 209(2), 171–176 (2004).
[CrossRef] [PubMed]

Kissel, V. N.

A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed-material plate,” Phys. Rev. Lett. 92(7), 077401 (2004).
[CrossRef] [PubMed]

Lagarkov, A. N.

A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed-material plate,” Phys. Rev. Lett. 92(7), 077401 (2004).
[CrossRef] [PubMed]

Lock, J. A.

Maheu, B.

B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).

Marston, P. L.

P. L. Marston and J. H. Crichton, “Radiation torque on a sphere caused by a circularly-polarized electromagnetic beam,” Phys. Rev. A 30(5), 2508–2516 (1984).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik Trüber Medien, Speziell Kolloidaler Metallösungen,” Ann. Phys. 330(3), 377–445 (1908).
[CrossRef]

Miroshnichenko, A. E.

A. E. Miroshnichenko, “Non-Rayleigh limit of the Lorenz-Mie solution and suppression of scattering by spheres of negative refractive index,” Phys. Rev. A 80(1), 013808 (2009).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

O’Neal, D. P.

D. P. O’Neal, L. R. Hirsch, N. J. Halas, J. D. Payne, and J. L. West, “Photo-thermal tumor ablation in mice using near infrared-absorbing nanoparticles,” Cancer Lett. 209(2), 171–176 (2004).
[CrossRef] [PubMed]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Payne, J. D.

D. P. O’Neal, L. R. Hirsch, N. J. Halas, J. D. Payne, and J. L. West, “Photo-thermal tumor ablation in mice using near infrared-absorbing nanoparticles,” Cancer Lett. 209(2), 171–176 (2004).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Polaert, H.

H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. 155(1-3), 169–179 (1998).
[CrossRef]

Ren, K. F.

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized lorenz-mie theory,” Appl. Opt. 37(19), 4218–4225 (1998).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994).
[CrossRef]

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Sibbett, W.

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66(6), 063402 (2002).
[CrossRef]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
[CrossRef]

Vier, D. C.

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

Ann. Phys. (1)

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[CrossRef]

Appl. Opt. (3)

Biomed. Opt. Express (1)

Cancer Lett. (1)

D. P. O’Neal, L. R. Hirsch, N. J. Halas, J. D. Payne, and J. L. West, “Photo-thermal tumor ablation in mice using near infrared-absorbing nanoparticles,” Cancer Lett. 209(2), 171–176 (2004).
[CrossRef] [PubMed]

IEEE Trans. Microw. Theory Tech. (1)

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[CrossRef]

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K. Volke-Sepulveda, V. Garcés-Chavez, S. Chávez-Cerda, J. Arlt, and D. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002).
[CrossRef]

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[CrossRef]

Phys. Rev. B (1)

Z. Ye, “Optical transmission and reflection of perfect lenses by left handed materials,” Phys. Rev. B 67(19), 193106 (2003).
[CrossRef]

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A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005).
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Figures (7)

Fig. 1
Fig. 1

Tx profile for an x-polarized TEM00 laser beam (λ = 384 nm, ω 0 = 3.7 μm) impinging on NRI or PRI particles with a = 7.5 μm and |Nim | = 10−7 and different Nr . (a) and (b) (x 0,z 0) = (0,0); (c) and (d) (x 0,z 0) = (0,150) μm. In (c) and (d), the curves for |Nr | = 2.00 and 1.50 are almost superposed and indistinguishable. The summation in (1) for these parameters leads to approximate the same slopes for Nr > 0 and Nr < 0. The refractive index of the host medium has been normalized to M = 1.00.

Fig. 2
Fig. 2

Tx profile for an x-polarized TEM00 laser beam (λ = 1064 nm, ω 0 = 3.7 μm) and a spherical particle with a = 0.75 μm and |Nim | = 10−7. The position of the particle is (x 0,y 0,z 0) = (0,1.9,0) μm. Only the first 10 an ’s and bn ’s significantly contribute to the torque profiles. Resonances in the Mie coefficients are reflected in the peaks observed.

Fig. 3
Fig. 3

Tx profile as function of Nim for (a) and (b): a PRI particle; (c) and (d) a NRI particle. The electromagnetic parameters of the beam and the location of particle are the same as in Fig. 2. Different resonances in the Mie scattering coefficients lead to different torque profiles.

Fig. 4
Fig. 4

Tz profile (due to circular polarization) for (a) PRI and (c) NRI particles with a = 7.5 μm, Nim = 10−7 and different Nr . The associated radiation pressure cross sections Cpr , z are shown in (b) and (d), respectively. In (b), the undefined region −2.8x10−4 m < z 0 < −1.0x10−4 m for Nr = 1.005 represents negative Cpr , z not shown due to the logarithmic scale. Beam parameters are λ = 384 nm and ω 0 = 3.7 μm.

Fig. 5
Fig. 5

Same as Fig. 4, using the same parameters for both the laser beam and the particle as those of Fig. 2.

Fig. 6
Fig. 6

Tz profile for an right-hand circularly polarized TEM00 laser beam (λ = 1064 nm, w = 3.7 μm) and a spherical particle with a = 0.75 μm located at (x 0,y 0,z 0) = (0,0,0). Only the first 10 an and bn significantly contribute to the torque profiles. Resonances in the Mie coefficients are reflected in the peaks observed.

Fig. 7
Fig. 7

(a) Tz profile for an right-hand circularly polarized TEM00 laser beam (λ = 1064 nm, w = 3.7 μm) and a NRI spherical particle with a = 0.75 μm located at (x 0,y 0,z 0) = (0,0,0). (b), (c) and (d) show the Mie scattering coefficients responsible for the peaks observed in (a).

Equations (3)

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T x = 2 M c π k 3 p = 1 n = p 2 n + 1 n ( n + 1 ) ( n + p ) ! ( n p ) ! Re { ( g n , T M p 1 g n , T M p , g n , T M p g n , T M p + 1 , ) [ 2 | a n | 2 ( a n + a n ) ] + ( g n , T E p 1 g n , T E p , g n , T E p g n , T E p + 1 , ) [ 2 | b n | 2 ( b n + b n ) ] } ,
T y = 2 M c π k 3 p = 1 n = p 2 n + 1 n ( n + 1 ) ( n + p ) ! ( n p ) ! Im { ( g n , T M p 1 g n , T M p , g n , T M p g n , T M p + 1 , ) [ 2 | a n | 2 ( a n + a n ) ] + ( g n , T E p 1 g n , T E p , g n , T E p g n , T E p + 1 , ) [ 2 | b n | 2 ( b n + b n ) ] } ,
T z = 4 M c π k 3 p = n = | p 0 | p 2 n + 1 n ( n + 1 ) ( n + | p | ) ! ( n | p | ) ! [ | g n , T M p | 2 ( Re ( a n ) | a n | 2 ) + | g n , T E p | 2 ( Re ( b n ) | b n | 2 ) ] ,

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