Abstract

Based on the generalized Lorentz–Mie theory (GLMT) and the localized approximation of the beam shape coefficients, we derived the expansions of incident elliptic Gaussian (EG) beams in terms of spherical vector wave functions (SVWFs). Utilizing multiple scattering (MS) equations and electromagnetic momentum (EM) theory, the lateral binding force (BF) exerted on a bi-sphere induced by an EG beam is calculated. Numerical effects of various parameters such as beam waist widths, beam polarization states, incident wavelengths, particle sizes, and material losses are analyzed and compared with the results of a circular Gaussian (CG) beam in detail. The observed dependence of the separation of optically bound particles on the incidence of an EG beam is in agreement with earlier theoretical predictions. Accurate investigation of BF induced by an EG beam could provide an effective test for further research on BF between more complex particles, which plays an important role in using optical manipulation on particle self-assembly.

© 2018 Optical Society of America

Full Article  |  PDF Article
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References

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  6. D. Maystre and P. Vincent, “Phenomenological study of binding in optically trapped photonic crystals,” J. Opt. Soc. Am. A 24, 2383–2393 (2007).
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    [Crossref]
  12. M. Nietovesperinas, J. J. Sáenz, R. Gómezmedina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 11428 (2010).
    [Crossref]
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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]
  22. K. F. Ren, G. Grehan, and G. Gouesbet, “Electromagnetic field expression of a laser sheet and the order of approximation,” J. Opt. 25, 165–176 (1994).
    [Crossref]
  23. D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 73, 113–124 (1997).
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    [Crossref]
  25. K. F. Ren, G. Grehan, and G. Gouesbet, “Localized approximation of generalized Lorenz-Mie theory: faster algorithm for computations of beam shape coefficient,” Part. Part. Syst. Charact. 9, 144–150 (1992).
    [Crossref]
  26. Z. J. Li, Z. S. Wu, and H. Y. Li, “Analysis of electromagnetic scattering by uniaxial anisotropic bispheres,” J. Opt. Soc. Am. A. 28, 118–125 (2011).
    [Crossref]
  27. Y. L. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996).
    [Crossref]
  28. Y. L. Xu and B. Å. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
    [Crossref]
  29. C. F. Bohren and D. R. Huffman, “Absorption and scattering of light by small particles,” Opt. Laser Technol. 31, 328 (1999).
    [Crossref]
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    [Crossref]
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    [Crossref]
  33. P. C. Chaumet and M. Nieto-Vesperinas, “Optical binding of particles with or without the presence of a flat dielectric surface,” Phys. Rev. B 64, 314–319 (2001).
    [Crossref]
  34. P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Selective nanomanipulation using optical forces,” Phys. Rev. B 66, 248 (2007).
  35. V. Demergis and E. L. Florin, “Ultrastrong optical binding of metallic nanoparticles,” Nano Lett. 12, 5756–5760 (2012).
    [Crossref]

2014 (1)

P. C. Chaumet and A. Rahmani, “Optical binding of magnetodielectric Rayleigh particles,” Phys. Rev. B 87, 2746–2752 (2014).
[Crossref]

2012 (1)

V. Demergis and E. L. Florin, “Ultrastrong optical binding of metallic nanoparticles,” Nano Lett. 12, 5756–5760 (2012).
[Crossref]

2011 (1)

Z. J. Li, Z. S. Wu, and H. Y. Li, “Analysis of electromagnetic scattering by uniaxial anisotropic bispheres,” J. Opt. Soc. Am. A. 28, 118–125 (2011).
[Crossref]

2010 (4)

T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 526 (2010).
[Crossref]

K. Dholakia and P. Zemánek, “Colloquium: gripped by light: optical binding,” Rev. Mod. Phys. 82, 1767–1791 (2010).
[Crossref]

K. Jay, P. C. Chaumet, T. N. Langtry, and A. Rahmani, “Optical binding of electrically small magnetodielectric particles,” J. Nanophoton. 4, 041583 (2010).
[Crossref]

M. Nietovesperinas, J. J. Sáenz, R. Gómezmedina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 11428 (2010).
[Crossref]

2007 (2)

D. Maystre and P. Vincent, “Phenomenological study of binding in optically trapped photonic crystals,” J. Opt. Soc. Am. A 24, 2383–2393 (2007).
[Crossref]

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Selective nanomanipulation using optical forces,” Phys. Rev. B 66, 248 (2007).

2006 (2)

Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568–570 (2006).
[Crossref]

T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, “Stable optical trapping based on optical binding forces,” Phys. Rev. Lett. 96, 113903 (2006).
[Crossref]

2005 (1)

J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters,” Phys. Rev. B 72, 1–4 (2005).
[Crossref]

2004 (2)

D. McGloin, A. E. Carruthers, K. Dholakia, and E. M. Wright, “Optically bound microscopic particles in one dimension,” Phys. Rev. E 69, 372–375 (2004).
[Crossref]

M. Mckendry, H. Mcgloin, D. Saberi, L. Caudwell, A. R. Brady, and M. Singer, “Randomised controlled trial assessing the impact of a nurse delivered, flow monitored protocol for optimisation of circulatory status after cardiac surgery,” Brit. Med. J. 258329, 438 (2004).

2003 (1)

2002 (2)

S. C. Dakin and P. J. Bex, “Role of synchrony in contour binding: some transient doubts sustained,” J. Opt. Soc. Am. A 19, 678–686 (2002).
[Crossref]

H. Xu and M. Käll, “Surface-plasmon-enhanced optical forces in silver nanoaggregates,” Phys. Rev. Lett. 89, 246802 (2002).
[Crossref]

2001 (2)

P. C. Chaumet and M. Nieto-Vesperinas, “Optical binding of particles with or without the presence of a flat dielectric surface,” Phys. Rev. B 64, 314–319 (2001).
[Crossref]

Y. L. Xu and B. Å. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
[Crossref]

1999 (1)

C. F. Bohren and D. R. Huffman, “Absorption and scattering of light by small particles,” Opt. Laser Technol. 31, 328 (1999).
[Crossref]

1997 (1)

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 73, 113–124 (1997).

1996 (1)

Y. L. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996).
[Crossref]

1995 (1)

Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres: asymmetry parameter,” Phys. Lett. A 11, 30–36 (1995).

1994 (2)

K. F. Ren, G. Grehan, and G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz-Mie theory by using a localized approximation,” J. Opt. Soc. Am. A. 11, 2072–2079 (1994).
[Crossref]

K. F. Ren, G. Grehan, and G. Gouesbet, “Electromagnetic field expression of a laser sheet and the order of approximation,” J. Opt. 25, 165–176 (1994).
[Crossref]

1993 (1)

A. A. Naqwi, “Performance prediction of dual cylindrical wave laser devices for flow measurements,” Exp. Fluids 14, 121–132 (1993).
[Crossref]

1992 (1)

K. F. Ren, G. Grehan, and G. Gouesbet, “Localized approximation of generalized Lorenz-Mie theory: faster algorithm for computations of beam shape coefficient,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[Crossref]

1990 (1)

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical matter: crystallization and binding in intense optical fields,” Science 249, 749–754 (1990).
[Crossref]

1989 (2)

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical binding,” Phys. Rev. Lett. 63, 1233–1236 (1989).
[Crossref]

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, 4594–4962 (1989).
[Crossref]

1980 (1)

1979 (1)

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

1977 (1)

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189–194 (1977).
[Crossref]

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[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, 4594–4962 (1989).
[Crossref]

Andrews, D. L.

T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 526 (2010).
[Crossref]

Ashkin, A.

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[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, 4594–4962 (1989).
[Crossref]

Bernet, S.

Bex, P. J.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, “Absorption and scattering of light by small particles,” Opt. Laser Technol. 31, 328 (1999).
[Crossref]

Brady, A. R.

M. Mckendry, H. Mcgloin, D. Saberi, L. Caudwell, A. R. Brady, and M. Singer, “Randomised controlled trial assessing the impact of a nurse delivered, flow monitored protocol for optimisation of circulatory status after cardiac surgery,” Brit. Med. J. 258329, 438 (2004).

Burns, M. M.

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical matter: crystallization and binding in intense optical fields,” Science 249, 749–754 (1990).
[Crossref]

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical binding,” Phys. Rev. Lett. 63, 1233–1236 (1989).
[Crossref]

Cai, Y.

Carruthers, A. E.

D. McGloin, A. E. Carruthers, K. Dholakia, and E. M. Wright, “Optically bound microscopic particles in one dimension,” Phys. Rev. E 69, 372–375 (2004).
[Crossref]

Caudwell, L.

M. Mckendry, H. Mcgloin, D. Saberi, L. Caudwell, A. R. Brady, and M. Singer, “Randomised controlled trial assessing the impact of a nurse delivered, flow monitored protocol for optimisation of circulatory status after cardiac surgery,” Brit. Med. J. 258329, 438 (2004).

Chan, C. T.

J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters,” Phys. Rev. B 72, 1–4 (2005).
[Crossref]

Chantada, L.

Chaumet, P. C.

P. C. Chaumet and A. Rahmani, “Optical binding of magnetodielectric Rayleigh particles,” Phys. Rev. B 87, 2746–2752 (2014).
[Crossref]

K. Jay, P. C. Chaumet, T. N. Langtry, and A. Rahmani, “Optical binding of electrically small magnetodielectric particles,” J. Nanophoton. 4, 041583 (2010).
[Crossref]

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Selective nanomanipulation using optical forces,” Phys. Rev. B 66, 248 (2007).

P. C. Chaumet and M. Nieto-Vesperinas, “Optical binding of particles with or without the presence of a flat dielectric surface,” Phys. Rev. B 64, 314–319 (2001).
[Crossref]

Cižmár, T.

T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 526 (2010).
[Crossref]

Dakin, S. C.

Dávila Romero, L. C.

T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 526 (2010).
[Crossref]

Davis, L. W.

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

Demergis, V.

V. Demergis and E. L. Florin, “Ultrastrong optical binding of metallic nanoparticles,” Nano Lett. 12, 5756–5760 (2012).
[Crossref]

Dholakia, K.

K. Dholakia and P. Zemánek, “Colloquium: gripped by light: optical binding,” Rev. Mod. Phys. 82, 1767–1791 (2010).
[Crossref]

T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 526 (2010).
[Crossref]

D. McGloin, A. E. Carruthers, K. Dholakia, and E. M. Wright, “Optically bound microscopic particles in one dimension,” Phys. Rev. E 69, 372–375 (2004).
[Crossref]

Florin, E. L.

V. Demergis and E. L. Florin, “Ultrastrong optical binding of metallic nanoparticles,” Nano Lett. 12, 5756–5760 (2012).
[Crossref]

Fournier, J. M.

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical matter: crystallization and binding in intense optical fields,” Science 249, 749–754 (1990).
[Crossref]

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical binding,” Phys. Rev. Lett. 63, 1233–1236 (1989).
[Crossref]

Frick, M.

Golovchenko, J. A.

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical matter: crystallization and binding in intense optical fields,” Science 249, 749–754 (1990).
[Crossref]

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical binding,” Phys. Rev. Lett. 63, 1233–1236 (1989).
[Crossref]

Gómezmedina, R.

Gouesbet, G.

K. F. Ren, G. Grehan, and G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz-Mie theory by using a localized approximation,” J. Opt. Soc. Am. A. 11, 2072–2079 (1994).
[Crossref]

K. F. Ren, G. Grehan, and G. Gouesbet, “Electromagnetic field expression of a laser sheet and the order of approximation,” J. Opt. 25, 165–176 (1994).
[Crossref]

K. F. Ren, G. Grehan, and G. Gouesbet, “Localized approximation of generalized Lorenz-Mie theory: faster algorithm for computations of beam shape coefficient,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[Crossref]

Grehan, G.

K. F. Ren, G. Grehan, and G. Gouesbet, “Electromagnetic field expression of a laser sheet and the order of approximation,” J. Opt. 25, 165–176 (1994).
[Crossref]

K. F. Ren, G. Grehan, and G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz-Mie theory by using a localized approximation,” J. Opt. Soc. Am. A. 11, 2072–2079 (1994).
[Crossref]

K. F. Ren, G. Grehan, and G. Gouesbet, “Localized approximation of generalized Lorenz-Mie theory: faster algorithm for computations of beam shape coefficient,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[Crossref]

Grzegorczyk, T. M.

T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, “Stable optical trapping based on optical binding forces,” Phys. Rev. Lett. 96, 113903 (2006).
[Crossref]

Gustafson, B. Å. S.

Y. L. Xu and B. Å. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
[Crossref]

Halas, N. J.

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 73, 113–124 (1997).

He, S.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, “Absorption and scattering of light by small particles,” Opt. Laser Technol. 31, 328 (1999).
[Crossref]

Jay, K.

K. Jay, P. C. Chaumet, T. N. Langtry, and A. Rahmani, “Optical binding of electrically small magnetodielectric particles,” J. Nanophoton. 4, 041583 (2010).
[Crossref]

Käll, M.

H. Xu and M. Käll, “Surface-plasmon-enhanced optical forces in silver nanoaggregates,” Phys. Rev. Lett. 89, 246802 (2002).
[Crossref]

Kemp, B. A.

T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, “Stable optical trapping based on optical binding forces,” Phys. Rev. Lett. 96, 113903 (2006).
[Crossref]

Kong, J. A.

T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, “Stable optical trapping based on optical binding forces,” Phys. Rev. Lett. 96, 113903 (2006).
[Crossref]

Langtry, T. N.

K. Jay, P. C. Chaumet, T. N. Langtry, and A. Rahmani, “Optical binding of electrically small magnetodielectric particles,” J. Nanophoton. 4, 041583 (2010).
[Crossref]

Li, H. Y.

Z. J. Li, Z. S. Wu, and H. Y. Li, “Analysis of electromagnetic scattering by uniaxial anisotropic bispheres,” J. Opt. Soc. Am. A. 28, 118–125 (2011).
[Crossref]

Li, R.

N. Ma and R. Li, “Analysis of radiation pressure force exerted on a sphere induced by laser sheet using Debye series,” in International Symposium on Antennas Propagation and Em Theory (IEEE, 2010), pp. 646–649.

Li, Z. J.

Z. J. Li, Z. S. Wu, and H. Y. Li, “Analysis of electromagnetic scattering by uniaxial anisotropic bispheres,” J. Opt. Soc. Am. A. 28, 118–125 (2011).
[Crossref]

Lin, Z.

J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters,” Phys. Rev. B 72, 1–4 (2005).
[Crossref]

Ma, N.

N. Ma and R. Li, “Analysis of radiation pressure force exerted on a sphere induced by laser sheet using Debye series,” in International Symposium on Antennas Propagation and Em Theory (IEEE, 2010), pp. 646–649.

Maystre, D.

McGloin, D.

D. McGloin, A. E. Carruthers, K. Dholakia, and E. M. Wright, “Optically bound microscopic particles in one dimension,” Phys. Rev. E 69, 372–375 (2004).
[Crossref]

Mcgloin, H.

M. Mckendry, H. Mcgloin, D. Saberi, L. Caudwell, A. R. Brady, and M. Singer, “Randomised controlled trial assessing the impact of a nurse delivered, flow monitored protocol for optimisation of circulatory status after cardiac surgery,” Brit. Med. J. 258329, 438 (2004).

Mckendry, M.

M. Mckendry, H. Mcgloin, D. Saberi, L. Caudwell, A. R. Brady, and M. Singer, “Randomised controlled trial assessing the impact of a nurse delivered, flow monitored protocol for optimisation of circulatory status after cardiac surgery,” Brit. Med. J. 258329, 438 (2004).

Naqwi, A. A.

A. A. Naqwi, “Performance prediction of dual cylindrical wave laser devices for flow measurements,” Exp. Fluids 14, 121–132 (1993).
[Crossref]

Ng, J.

J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters,” Phys. Rev. B 72, 1–4 (2005).
[Crossref]

Nietovesperinas, M.

Nieto-Vesperinas, M.

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Selective nanomanipulation using optical forces,” Phys. Rev. B 66, 248 (2007).

P. C. Chaumet and M. Nieto-Vesperinas, “Optical binding of particles with or without the presence of a flat dielectric surface,” Phys. Rev. B 64, 314–319 (2001).
[Crossref]

Rahmani, A.

P. C. Chaumet and A. Rahmani, “Optical binding of magnetodielectric Rayleigh particles,” Phys. Rev. B 87, 2746–2752 (2014).
[Crossref]

K. Jay, P. C. Chaumet, T. N. Langtry, and A. Rahmani, “Optical binding of electrically small magnetodielectric particles,” J. Nanophoton. 4, 041583 (2010).
[Crossref]

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Selective nanomanipulation using optical forces,” Phys. Rev. B 66, 248 (2007).

Ren, K. F.

K. F. Ren, G. Grehan, and G. Gouesbet, “Electromagnetic field expression of a laser sheet and the order of approximation,” J. Opt. 25, 165–176 (1994).
[Crossref]

K. F. Ren, G. Grehan, and G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz-Mie theory by using a localized approximation,” J. Opt. Soc. Am. A. 11, 2072–2079 (1994).
[Crossref]

K. F. Ren, G. Grehan, and G. Gouesbet, “Localized approximation of generalized Lorenz-Mie theory: faster algorithm for computations of beam shape coefficient,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[Crossref]

Ritschmarte, M.

Roosen, G.

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189–194 (1977).
[Crossref]

Saberi, D.

M. Mckendry, H. Mcgloin, D. Saberi, L. Caudwell, A. R. Brady, and M. Singer, “Randomised controlled trial assessing the impact of a nurse delivered, flow monitored protocol for optimisation of circulatory status after cardiac surgery,” Brit. Med. J. 258329, 438 (2004).

Sáenz, J. J.

Sarkar, D.

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Figures (6)

Fig. 1.
Fig. 1.

Configuration of bi-sphere induced by EG beam: d, interparticle distance; θ, angle between polarization and bi-sphere orientation.

Fig. 2.
Fig. 2.

Comparisons of our results on normalized BF in the x direction by a degenerated EG beam with that of a plane wave incidence using the coupled dipole method: (a) θ=0° polarization; (b): θ=90° polarization.

Fig. 3.
Fig. 3.

Effects of the varying beam waist radii on lateral BF Fx as a function of the inter-particle distance d induced by EG beam: (a) varying w0y; (b) varying w0x.

Fig. 4.
Fig. 4.

Comparisons of lateral BF acting on the right particle depending on the polarizations and incident wavelengths induced by EG beam with that by CG beam. (a) Incident wavelength λ=800  nm; (b) incident wavelength λ=600  nm; (c) incident wavelength λ=532  nm; (d) incident wavelength λ=400  nm.

Fig. 5.
Fig. 5.

Comparisons of lateral BF acting on the right particle depending on the particle sizes induced by EG beam with that by CG beam.

Fig. 6.
Fig. 6.

Comparisons of lateral BF acting on the right particle depending on the particle material losses induced by EG beam with that by CG beam.

Equations (44)

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2E(r,t)1c22t2E(r,t)=0.
{Ex(x,y,z)=E0ψ0sheikzx^Ey(x,y,z)=0Ez(x,y,z)=2QxxE0ψ0sheikzz^/hx,
{Hx(x,y,z)=0Hy(x,y,z)=H0ψ0sheikzy^Hz(x,y,z)=2QyytH0ψ0sheikzz^/hy,
ψ0sh=iQxQyexp(iQxx2/w0x2+iQyy2/w0y2),
Ejinc=E0n=1m=nn[ajmniNmn(1)+bjmniMmn(1)]Hjinc=E0kiωμn=1m=nn[ajmniMmn(1)+bjmniNmn(1)],
ajmni=Cnmgjn,TMmbjmni=iCnmgjn,TEm,
Cnm={in12n+1n(n+1),m0(1)min1(nm)!(n+m)!2n+1n(n+1),m<0,
(g¯jn,TM0g¯jn,TE0)=12Zn0(1i)exp(ikzj)(F^ψ00,sh)j=0q=0p=0q+[(j+1)/2]F^(Ap+qBjCj)p!q!j!(2q2p+j+1)!×[F^C2q2p+1+F^(B2q2p+1B2q2p+1)].
(g¯jn,TMmg¯jn,TEm)=12Znm(1i)exp(ikzj)(F^ψ00,sh)×{j=0q=0p=0q+[(m+j1)/2]F^(Ap+qB2q2p+j+m1Cj)p!q!j!(2q2p+j+m1)!+(11)j=0q=0p=0q+[(m+j+1)/2]F^(Ap+qB2q2p+j+m+1Cj)p!q!j!(2q2p+j+m+1)!}.
(g¯n,TM|m|g¯jn,TE|m|)=12Znm(1i)exp(ikzj)(F^ψ00,sh)×{j=0q=0p=0q+[(|m|+j1)/2]F^(Ap+qBjC2q2p+j+|m|1)p!q!j!(2q2p+j+|m|1)!+(11)j=0q=0p=0q+[(|m|+j+1)/2]F^(Ap+qBjC2q2p+j+|m|+1)p!q!j!(2q2p+j+|m|+1)!},
Znm={2n(n+1)2n+1i,m=0(2i2n+1)|m|1,m0,
ψ00,sh=Q0xQ0yexp[i2(Q0xw0x2+Q0yw0y2)r2sin2θ+i(Q0xw0x2x2+Q0yw0y2y2)],
A=r2sin2θ4(iQ0xw0x2+iQ0yw0y2),
B=rsinθ(iQ0xw0x2x+Q0yw0y2y),
C=rsinθ(iQ0xw0x2xQ0yw0y2y),
Q0x=1i+[2/(kw0x2)](rcosθz),
Q0y=1i+[2/(kw0y2)](rcosθz),
Ejs=E0n=1m=nn[ajmnsMmn(3)+bjmnsNmn(3)]Hjs=E0kiωμn=1m=nn[ajmnsNmn(3)+bjmnsMmn(3)],
Ej1=E0n=1m=nn[Ajmn1Nmn(1)+kkjμjμBjmn1Mmn(1)]Hj1=E0kiωμn=1m=nn[Bjmn1Nmn(1)+kjkμμjAjmn1Mmn(1)],
Ej1|t=Ejit|t+Ejs|t,Hj1|t=Hjit|t+Hjs|t,
Ejit=Ejinc+Ep,js,Hjit=Hjinc+Hp,js,
Ejit=E0n=1m=nn[ajmnitMmn(1)+bjmnitNmn(1)]Hjit=E0kiωμn=1m=nn[ajmnitNmn(1)+bjmnitMmn(1)],
ajmnit=ajmni+v=1μ=vv[apμvsAmnμv+bpμvsBmnμv](pj)bjmnit=bjmni+v=1μ=vv[apμvsBmnμv+bpμvsAmnμv](pj).
ajmnitjn(kr)+ajmnshn(1)(kr)=kkjμjμBjmn1jn(kjr)
bjmnitjn(kr)+bjmnshn(1)(kr)=kjkμμjAjmn1jn(kjr),
ajmnitddr[rjn(kr)]+ajmnsddr[rhn(1)(kr)]=Bjmn1kkjddr[rjn(kjr)],
bjmnitddr[rjn(kr)]+bjmnsddr[rhn(1)(kr)]=Ajmn1kkjddr[rjn(kjr)].
ajmns=ajnajmnit=ajn{ajmni+v=1μ=vv[apμvsAmnμv+bpμvsBmnμv](pj)}bjmns=bjnbjmnit=bjn{bjmni+v=1μ=vv[apμvsBmnμv+bpμvsAmnμv](pj)},
F=sn^·Tds,
T=12Re[ϵ0EE*+μ0HH*12ϵ0|E|2I12μ0|H|2I],
F=12Re02π0π[ϵ0EjrEj+μ0HjrHj12(ϵ0Ej2+μ0Hj2)e^r]r2sinθdθdϕ|r>a.
F=12Re02π0π12(ϵ0Ej2+μ0Hj2)e^rr2sinθdθdϕ|r>a.
F=12Re02π0π12[ϵ0(EjθitEjθit*+EjθitEjθs*+EjθsEjθit*+EjθsEjθs*+EjϕitEjϕs*+EjϕsEjϕi*+EjϕsEjϕs*+EjϕitEjϕit*)+μ0(HjθitHjθit*+HjθitHjθs*+HjθsHjθit*+HjθsHjθs*+HjϕitHjϕs*+HjϕsHjϕit*+HjϕsHjϕs*+HjϕitHjϕit*)]e^rr2sinθdθdϕ|r>a.
ϵ0Ejθs=μ0Hjϕsϵ0Ejϕs=μ0Hjθs.
e^r=sinθcosϕe^x+sinθsinϕe^y+cosθe^z.
Fx=n02cRe02π0π(EjϕitHjθs*EjθitHjϕs*+HjθitEjϕs*HjϕitEjθs*+EjϕsHjθs*EjθsHjϕs*)r2sin2θcosϕdθdϕ.
Fx=n0P0πck02w0xw0yRe{n=1m=nn[(nm)(n+m+1)Nmn1Nm+1n1(ajmnitbjm+1ns*+bjmnitajm+1ns*+ajmnsajm+1nit*+ajmnsbjm+1nit*+2ajmnsbjm+1ns*+2bjmnsajm+1ns*)i(nm1)(nm)(2n1)(2n+1)(n1)(n+1)Nmn1Nm+1n11(ajmnitajm+1n1s*+bjmnitbjm+1n1s*+ajmnsajm+1n1it*+bjmnsbjm+1n1it*+2ajmnsajm+1n1s*+2bjmnsbjm+1n1s*)i(n+m+1)(n+m+2)(2n+1)(2n+3)n(n+2)Nmn1Nm+1n+11(ajmnitajm+1n+1s*+bjmnitbjm+1n+1s*+ajmnsajm+1n+1it*+bjmnsbjm+1n+1it*+2ajmnsajm+1n+1s*+2bjmnsbjm+1n+1s*)]},
Nmn=(2n+1)(nm)!4π(n+m)!(m=0,±1,,±n).
a1jmnit=ajmni,b1jmnit=bjmni.
a1jmns=ajn1ajmnit,b1jmns=bjn1bjmnit,
aijmnit=ajmni+v=1μ=vv[ai1pμvsAmnμv+bi1pμvsBmnμv](lj)bijmnit=bjmni+v=1μ=vv[ai1pμvsBmnμv+bi1pμvsAmnμv](lj),
aipμvs=(1f)ai1pμvs+fapvaijmnit(0f1)bipμvs=(1f)bi1pμvs+fbpvbijmnit(0f1).
Ejθit=E0n=1m=nn[ajmnitjn(kr)imPnm(cosθ)sinθ+bjmnit1krd(rjn(kr))drdPnm(cosθ)dθ]eimϕEjϕit=E0n=1m=nn[ajmnitjn(kr)dPnm(cosθ)dθ+bjmnit1krd(rjn(kr))drimPnm(cosθ)sinθ]eimϕHjθit=k0E0iωμn=1m=nn[bjmnitjn(kr)imPnm(cosθ)sinθ+ajmnit1krd(rjn(kr))drdPnm(cosθ)dθ]eimϕHjϕit=k0E0iωμn=1m=nn[bjmnitjn(kr)dPnm(cosθ)dθ+ajmnit1krd(rjn(kr))drimPnm(cosθ)sinθ]eimϕEjθs=E0n=1m=nn[ajmnshn(1)(kr)imPnm(cosθ)sinθ+bjmns1krd(rhn(1)(kr))drdPnm(cosθ)dθ]eimϕEjϕs=E0n=1m=nn[ajmnshn(1)(kr)dPnm(cosθ)dθ+bjmns1krd(rhn(1)(kr))drimPnm(cosθ)sinθ]eimϕHjθs=k0E0iωμn=1m=nn[bjmnshn(1)(kr)imPnm(cosθ)sinθ+ajmns1krd(rhn(1)(kr))drdPnm(cosθ)dθ]eimϕHjϕs=k0E0iωμn=1m=nn[bjmnshn(1)(kr)dPnm(cosθ)dθ+ajmns1krd(rhn(1)(kr))drimPnm(cosθ)sinθ]eimϕ,
02πei(mm)ϕdϕ0π[mdPnm(cosθ)dθPnm(cosθ)sinθ+mPnm(cosθ)sinθdPnm(cosθ)dθ]sin2θeiϕdθ=(nm)(n+m+1)Nmn1Nmn1δm+1,mδn,n02πei(mm)ϕdϕ0π[mdPnm(cosθ)dθPnm(cosθ)sinθ+mPnm(cosθsinθdPnm(cosθ)dθ]sin2θeiϕdθ=(nm)(n+m+1)Nmn1Nmn1δm+1,mδn,n02πei(mm)ϕdϕ0π[dPnm(cosθ)dθdPnm(cosθ)dθ+mmPnm(cosθ)sinθPnm(cosθ)sinθ]sin2θeiϕdθ=Nmn1Nmn1(n+m+1)(n+m+2)(2n+1)(2n+3)n(n+2)δm+1,mδn+1,nNmn1Nmn1(nm)(nm+1)(2n+1)(2n+3)n(n+2)δm+1,mδn,n+102πei(mm)ϕdϕ0π[dPnm(cosθ)dθdPnm(cosθ)dθ+mmPnm(cosθ)sinθPnm(cosθ)sinθ]sin2θeiϕdθ=Nmn1Nmn1(n+m+1)(n+m+2)(2n+1)(2n+3)n(n+2)δm+1,mδn+1,nNmn1Nmn1(nm)(nm+1)(2n+1)(2n+3)n(n+2)δm+1,mδn,n+102πei(mm)ϕdϕ0π[mdPnm(cosθ)dθPnm(cosθ)sinθ+mPnm(cosθ)sinθdPnm(cosθ)dθ]sinθcosθdθ=mNmn1Nmn1δm,mδn,n02πei(mm)ϕdϕ0π[dPnm(cosθ)dθdPnm(cosθ)dθ+mmPnm(cosθ)sinθPnm(cosθ)sinθ]sinθcosθdθ=Nmn1Nmn1(nm+1)(n+m+1)(2n+1)(2n+3)n(n+2)δm,mδn,n+1Nmn1Nmn1(nm)(nm+1)(2n+1)(2n+3)n(n+2)δm,mδn+1,n,