Abstract

Based on the generalized Lorenz-Mie theory and the Maxwell stress tensor approach we present the first rigorous full-wave solution of the optical forces acting on spherical microparticles immersed in a two-dimensional vector Airy beam beyond the paraxial approximation. The critical aspect lies in evaluating efficiently and accurately the partial wave expansion coefficients of the incident Airy beam, which are achieved by using the vector angular spectrum representation for a variety of polarizations. The optical field distributions are then simulated to show the self-accelerating and self-healing effects of the Airy beam. The dielectric and gold microparticles are shown to be trapped within the main lobe or the nearby side-lobes mostly by the transverse gradient optical force while driven forward along the parabolic trajectory of the Airy beam by the longitudinal scattering force. It is thus demonstrated theoretically that the vector Airy beam has the capability of precisely transporting both dielectric and metallic microparticles along the prespecified curved paths.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  7. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]

2017 (3)

Q. Ye and H. Lin, “On deriving the maxwell stress tensor method for calculating the optical force and torque on an object in harmonic electromagnetic fields,” Eur. J. Phys. 38, 045202 (2017).
[Crossref]

H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017).
[Crossref]

X. Yu, Q. Ye, H. Chen, S. Liu, and Z. Lin, “Simple algorithm for partial wave expansion of plasmonic and evanescent fields,” Opt. Express 25, 4201–4215 (2017).
[Crossref] [PubMed]

2016 (1)

Z. Zhao, W. Zang, and J. Tian, “Optical trapping and manipulation of Mie particles with Airy beam,” J. Opt. 18, 025607 (2016).
[Crossref]

2015 (2)

H. Chen, S. Liu, J. Zi, and Z. Lin, “Fano resonance-induced negative optical scattering force on plasmonic nanoparticles,” ACS Nano 9, 1926–1935 (2015).
[Crossref] [PubMed]

F. J. Rodríguez-Fortuño, N. Engheta, A. Martínez, and A. V. Zayats, “Lateral forces on circularly polarizable particles near a surface,” Nat. Commun. 6, 8799 (2015).
[Crossref] [PubMed]

2014 (2)

S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014).
[PubMed]

H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, “Tailoring azimuthal optical force on lossy chiral particles in Bessel beams,” Phys. Rev. A 90, 043850 (2014).
[Crossref]

2013 (2)

N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[Crossref]

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
[Crossref]

2012 (1)

D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[Crossref] [PubMed]

2011 (4)

W. Lu, J. Chen, Z. Lin, and S. Liu, “Driving a dielectric cylindrical particle with a one dimensional Airy beam: a rigorous full wave solution,” Prog. Electromagn. Res. 115, 409–422 (2011).
[Crossref]

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[Crossref]

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

Z. Zheng, B. Zhang, H. Chen, J. Ding, and H. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (2)

A. V. Novitsky and D. V. Novitsky, “Nonparaxial airy beams: role of evanescent waves,” Opt. Lett. 34, 3430–3432 (2009).
[Crossref] [PubMed]

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2009).
[Crossref]

2008 (1)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]

2007 (2)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[Crossref] [PubMed]

2006 (2)

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref] [PubMed]

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

2005 (2)

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937–1942 (2005).
[Crossref] [PubMed]

J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

2004 (1)

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

2002 (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

2000 (1)

A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6, 841–856 (2000).
[Crossref]

1996 (1)

K. Unnikrishnan and A. R. P. Rau, “Uniqueness of the Airy packet in quantum mechanics,” Am. J. Phys. 64, 1034–1035 (1996).
[Crossref]

1995 (1)

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

1970 (1)

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

Ashkin, A.

A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6, 841–856 (2000).
[Crossref]

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

A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers (World Scientific, 2006).
[Crossref]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

Barbosa, L. C.

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref] [PubMed]

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

Bhatia, V. K.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937–1942 (2005).
[Crossref] [PubMed]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons: New York, 1983).

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Cesar, C. L.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref] [PubMed]

Chan, C. T.

S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014).
[PubMed]

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

Chen, H.

X. Yu, Q. Ye, H. Chen, S. Liu, and Z. Lin, “Simple algorithm for partial wave expansion of plasmonic and evanescent fields,” Opt. Express 25, 4201–4215 (2017).
[Crossref] [PubMed]

H. Chen, S. Liu, J. Zi, and Z. Lin, “Fano resonance-induced negative optical scattering force on plasmonic nanoparticles,” ACS Nano 9, 1926–1935 (2015).
[Crossref] [PubMed]

H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, “Tailoring azimuthal optical force on lossy chiral particles in Bessel beams,” Phys. Rev. A 90, 043850 (2014).
[Crossref]

Z. Zheng, B. Zhang, H. Chen, J. Ding, and H. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
[Crossref] [PubMed]

Chen, H. J.

H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017).
[Crossref]

Chen, J.

N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[Crossref]

W. Lu, J. Chen, Z. Lin, and S. Liu, “Driving a dielectric cylindrical particle with a one dimensional Airy beam: a rigorous full wave solution,” Prog. Electromagn. Res. 115, 409–422 (2011).
[Crossref]

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2009).
[Crossref]

Cheng, H.

Christodoulides, D. N.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Cižmár, T.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[Crossref]

Cruz, C. H. B.

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref] [PubMed]

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

Dholakia, K.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Ding, J.

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Engheta, N.

F. J. Rodríguez-Fortuño, N. Engheta, A. Martínez, and A. V. Zayats, “Lateral forces on circularly polarizable particles near a surface,” Nat. Commun. 6, 8799 (2015).
[Crossref] [PubMed]

Fontes, A.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref] [PubMed]

Garcés-Chávez, V.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Gouesbet, G.

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie theories (Springer: Berlin, 2011). And references therein.
[Crossref]

Gréhan, G.

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie theories (Springer: Berlin, 2011). And references therein.
[Crossref]

Grier, D. G.

D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[Crossref] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Hansen, P. M.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937–1942 (2005).
[Crossref] [PubMed]

Harrit, N.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937–1942 (2005).
[Crossref] [PubMed]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2006).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons: New York, 1983).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed (John Wiley and Sons, 1999).

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Kivshar, Y. S.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
[Crossref]

Klein, A. E.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
[Crossref]

Lin, H.

Q. Ye and H. Lin, “On deriving the maxwell stress tensor method for calculating the optical force and torque on an object in harmonic electromagnetic fields,” Eur. J. Phys. 38, 045202 (2017).
[Crossref]

Lin, Z.

X. Yu, Q. Ye, H. Chen, S. Liu, and Z. Lin, “Simple algorithm for partial wave expansion of plasmonic and evanescent fields,” Opt. Express 25, 4201–4215 (2017).
[Crossref] [PubMed]

H. Chen, S. Liu, J. Zi, and Z. Lin, “Fano resonance-induced negative optical scattering force on plasmonic nanoparticles,” ACS Nano 9, 1926–1935 (2015).
[Crossref] [PubMed]

H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, “Tailoring azimuthal optical force on lossy chiral particles in Bessel beams,” Phys. Rev. A 90, 043850 (2014).
[Crossref]

N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[Crossref]

W. Lu, J. Chen, Z. Lin, and S. Liu, “Driving a dielectric cylindrical particle with a one dimensional Airy beam: a rigorous full wave solution,” Prog. Electromagn. Res. 115, 409–422 (2011).
[Crossref]

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2009).
[Crossref]

J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

Lin, Z. F.

H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017).
[Crossref]

Liu, S.

X. Yu, Q. Ye, H. Chen, S. Liu, and Z. Lin, “Simple algorithm for partial wave expansion of plasmonic and evanescent fields,” Opt. Express 25, 4201–4215 (2017).
[Crossref] [PubMed]

H. Chen, S. Liu, J. Zi, and Z. Lin, “Fano resonance-induced negative optical scattering force on plasmonic nanoparticles,” ACS Nano 9, 1926–1935 (2015).
[Crossref] [PubMed]

H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, “Tailoring azimuthal optical force on lossy chiral particles in Bessel beams,” Phys. Rev. A 90, 043850 (2014).
[Crossref]

N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[Crossref]

W. Lu, J. Chen, Z. Lin, and S. Liu, “Driving a dielectric cylindrical particle with a one dimensional Airy beam: a rigorous full wave solution,” Prog. Electromagn. Res. 115, 409–422 (2011).
[Crossref]

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2009).
[Crossref]

Liu, S. Y.

H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017).
[Crossref]

Lu, W.

H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, “Tailoring azimuthal optical force on lossy chiral particles in Bessel beams,” Phys. Rev. A 90, 043850 (2014).
[Crossref]

W. Lu, J. Chen, Z. Lin, and S. Liu, “Driving a dielectric cylindrical particle with a one dimensional Airy beam: a rigorous full wave solution,” Prog. Electromagn. Res. 115, 409–422 (2011).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
[Crossref]

Martínez, A.

F. J. Rodríguez-Fortuño, N. Engheta, A. Martínez, and A. V. Zayats, “Lateral forces on circularly polarizable particles near a surface,” Nat. Commun. 6, 8799 (2015).
[Crossref] [PubMed]

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]

McGloin, D.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Melville, H.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Minovich, A. E.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
[Crossref]

Neshev, D. N.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
[Crossref]

Neves, A. A. R.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref] [PubMed]

Ng, J.

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2009).
[Crossref]

J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

Novitsky, A. V.

Novitsky, D. V.

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2006).
[Crossref]

Oddershede, L.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937–1942 (2005).
[Crossref] [PubMed]

Padilha, L. A.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref] [PubMed]

Pertsch, T.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
[Crossref]

Rau, A. R. P.

K. Unnikrishnan and A. R. P. Rau, “Uniqueness of the Airy packet in quantum mechanics,” Am. J. Phys. 64, 1034–1035 (1996).
[Crossref]

Rodriguez, E.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref] [PubMed]

Rodríguez-Fortuño, F. J.

F. J. Rodríguez-Fortuño, N. Engheta, A. Martínez, and A. V. Zayats, “Lateral forces on circularly polarizable particles near a surface,” Nat. Commun. 6, 8799 (2015).
[Crossref] [PubMed]

Ruffner, D. B.

D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[Crossref] [PubMed]

Sheng, P.

J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

Shi, L.

H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017).
[Crossref]

Sibbett, W.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Siviloglou, G. A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[Crossref] [PubMed]

Temme, N. M.

N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics (John Wiley and Sons, 1996).
[Crossref]

Tian, J.

Z. Zhao, W. Zang, and J. Tian, “Optical trapping and manipulation of Mie particles with Airy beam,” J. Opt. 18, 025607 (2016).
[Crossref]

H. Cheng, W. Zang, W. Zhou, and J. Tian, “Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam,” Opt. Express 18, 20384–20394 (2010).
[Crossref] [PubMed]

Unnikrishnan, K.

K. Unnikrishnan and A. R. P. Rau, “Uniqueness of the Airy packet in quantum mechanics,” Am. J. Phys. 64, 1034–1035 (1996).
[Crossref]

Wang, H.

Wang, N.

H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, “Tailoring azimuthal optical force on lossy chiral particles in Bessel beams,” Phys. Rev. A 90, 043850 (2014).
[Crossref]

N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[Crossref]

Wang, S. B.

S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014).
[PubMed]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
[Crossref]

Xu, Y.

Ye, Q.

Q. Ye and H. Lin, “On deriving the maxwell stress tensor method for calculating the optical force and torque on an object in harmonic electromagnetic fields,” Eur. J. Phys. 38, 045202 (2017).
[Crossref]

H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017).
[Crossref]

X. Yu, Q. Ye, H. Chen, S. Liu, and Z. Lin, “Simple algorithm for partial wave expansion of plasmonic and evanescent fields,” Opt. Express 25, 4201–4215 (2017).
[Crossref] [PubMed]

Yu, X.

Zang, W.

Z. Zhao, W. Zang, and J. Tian, “Optical trapping and manipulation of Mie particles with Airy beam,” J. Opt. 18, 025607 (2016).
[Crossref]

H. Cheng, W. Zang, W. Zhou, and J. Tian, “Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam,” Opt. Express 18, 20384–20394 (2010).
[Crossref] [PubMed]

Zangwill, A.

A. Zangwill, Modern Electrodynamics (Cambridge University Press, 2012).

Zayats, A. V.

F. J. Rodríguez-Fortuño, N. Engheta, A. Martínez, and A. V. Zayats, “Lateral forces on circularly polarizable particles near a surface,” Nat. Commun. 6, 8799 (2015).
[Crossref] [PubMed]

Zhan, Q.

Zhang, B.

Zhang, Y. W.

H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017).
[Crossref]

Zhao, Z.

Z. Zhao, W. Zang, and J. Tian, “Optical trapping and manipulation of Mie particles with Airy beam,” J. Opt. 18, 025607 (2016).
[Crossref]

Zheng, Z.

Zhou, W.

Zi, J.

H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017).
[Crossref]

H. Chen, S. Liu, J. Zi, and Z. Lin, “Fano resonance-induced negative optical scattering force on plasmonic nanoparticles,” ACS Nano 9, 1926–1935 (2015).
[Crossref] [PubMed]

ACS Nano (1)

H. Chen, S. Liu, J. Zi, and Z. Lin, “Fano resonance-induced negative optical scattering force on plasmonic nanoparticles,” ACS Nano 9, 1926–1935 (2015).
[Crossref] [PubMed]

Am. J. Phys. (2)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

K. Unnikrishnan and A. R. P. Rau, “Uniqueness of the Airy packet in quantum mechanics,” Am. J. Phys. 64, 1034–1035 (1996).
[Crossref]

Appl. Opt. (2)

Eur. J. Phys. (1)

Q. Ye and H. Lin, “On deriving the maxwell stress tensor method for calculating the optical force and torque on an object in harmonic electromagnetic fields,” Eur. J. Phys. 38, 045202 (2017).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6, 841–856 (2000).
[Crossref]

J. Opt. (1)

Z. Zhao, W. Zang, and J. Tian, “Optical trapping and manipulation of Mie particles with Airy beam,” J. Opt. 18, 025607 (2016).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Phys. A: Math. Gen. (1)

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

Laser Photon. Rev. (1)

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
[Crossref]

Nano Lett. (1)

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937–1942 (2005).
[Crossref] [PubMed]

Nat. Commun. (2)

S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014).
[PubMed]

F. J. Rodríguez-Fortuño, N. Engheta, A. Martínez, and A. V. Zayats, “Lateral forces on circularly polarizable particles near a surface,” Nat. Commun. 6, 8799 (2015).
[Crossref] [PubMed]

Nat. Photonics (3)

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]

Nature (2)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. A (3)

H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, “Tailoring azimuthal optical force on lossy chiral particles in Bessel beams,” Phys. Rev. A 90, 043850 (2014).
[Crossref]

N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[Crossref]

H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017).
[Crossref]

Phys. Rev. B (2)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

Phys. Rev. E (1)

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2009).
[Crossref]

Phys. Rev. Lett. (4)

D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[Crossref] [PubMed]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

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

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Prog. Electromagn. Res. (1)

W. Lu, J. Chen, Z. Lin, and S. Liu, “Driving a dielectric cylindrical particle with a one dimensional Airy beam: a rigorous full wave solution,” Prog. Electromagn. Res. 115, 409–422 (2011).
[Crossref]

Other (8)

N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics (John Wiley and Sons, 1996).
[Crossref]

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons: New York, 1983).

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie theories (Springer: Berlin, 2011). And references therein.
[Crossref]

J. D. Jackson, Classical Electrodynamics, 3rd ed (John Wiley and Sons, 1999).

A. Zangwill, Modern Electrodynamics (Cambridge University Press, 2012).

A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers (World Scientific, 2006).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
[Crossref]

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2006).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1

Self-accelerating and self-healing effects of a linear polarized Airy beam with oscillation along x-axis and wx = wy = 2λ. The field profiles |E|2 corresponding to (a) the incident 2D Airy beam and (b) the 2D Airy beam perturbed by a spherical particle with the radius R = λ, the permittivity εr = 2.53, and the location at (−1.8λ, −1.8λ, 10λ), (c) trajectories of the maximum intensity within the main lobe for both the non-paraxial and paraxial Airy beams. For comparison, trajectories for strongly non-paraxial Airy beam with wx = wy = λ are plotted in (c) with dashed lines. Panels (a) and (b) are composed of six transverse sections (constant z planes) and a longitudinal section (x = y plane).

Fig. 2
Fig. 2

Self-healing effect manifested by observing the field profiles |E|2 of the 2D Airy beam perturbed by a scatterer, along the transverse line y = x at three different propagation distances z = 15λ, 40λ, and 80λ, as indicated by the red solid line. The field profiles of the incident 2D Airy beam are shown as well for the convenience of the comparison, as indicated by the black solid line. All the parameters are the same as those in Fig. 1(b).

Fig. 3
Fig. 3

Optical forces versus the positions of a dielectric particle under the illumination of a linearly polarized Airy beam with polarization along x-axis, where the maximum intensity is normalized to 1 mWµm−2 in the initial plane z = 0 within the main lobe. The dielectric particle with the radius R = λ and the permittivity εr = 2.53 is located within the transverse plane z = 10λ. The map of the longitudinal optical force Fz (a) and the transverse optical force F (b) in this transverse plane are potted, where the thick arrows with red color denote directions and magnitudes ( F x 2 + F y 2 ) 1 / 2 of the transverse optical forces via the directions and lengths of the arrows. The transverse optical forces with |F| < 0.1 pN are not shown due to the negligible small magnitude. And the light gray arrows are the stream lines of the transverse optical forces to only indicate the directions.

Fig. 4
Fig. 4

Optical forces versus the positions of a gold nanoparticle suspended in water with the radius R = λ/20 and positioned on the transverse plane z = 10λ under the illumination of a linearly polarized Airy beam polarization along x-axis, where the maximum intensity is normalized to 10 mWµm−2 in the initial plane z = 0 within the main lobe. The dielectric function of the gold particle is εr = −48.45 + 3.6i at wavelength λ = 1.064 µm and the refractive index of water is 1.33. The map of the longitudinal optical forces Fz (a) and the transverse optical forces F (b) are plotted within this transverse plane, without showing |F| < 10 fN due to negligible small magnitude. And the light gray arrows denote the stream lines of transverse optical forces to only indicate the directions.

Fig. 5
Fig. 5

The maps of normalized optical trapping potential for the dielectric particle (a) and the gold nanoparticle (b) within the transverse plane z = 10λ, which are calculated based on the path integral of the transverse optical forces F. The parameters of the dielectric particle and the gold particle are the same as those in Fig. 3 and Fig. 4.

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

E inc ( x , y , z ) = A ( k x , k y ; z b ) e i [ k x x + k y y + k z ( z z b ) ] d k x d k y ,
E ( x , y , z b ) = E 0 Ai ( x x b w x ) Ai ( y y b w y ) exp ( α x x x b w x + α y y y b w y ) p ( x , y ) ,
A ( k x , k y ; z b ) = E 0 4 π 2 w x w y e 1 3 ( α x i k x w x ) 3 e 1 3 ( α y i k y w y ) 3 e i ( k x x b + k y y b ) s ( α , β ) ,
A = E 0 4 π 2 w x w y e 1 3 ( α x i k x w x ) 3 e 1 3 ( α y i k y w y ) 3 e i ( k x x b + k y y b ) s ( α , β ) ,
E inc ( r , θ , ϕ ) = n , m i E m n [ p m n N n m ( 1 ) ( k , r ) + q m n M n m ( 1 ) ( k , r ) ] ,
E sca ( r , θ , ϕ ) = n , m i E m n [ a m n N n m ( 3 ) ( k , r ) + b m n M n m ( 3 ) ( k , r ) ] ,
a m n = a n p m n , b m n = b n q m n .
F = S r ^ T d S ,
T = 1 2 Re [ ε E E + μ H H 1 2 ( ε E E + μ H H ) I ] ,
F x = Re [ F 1 ] , F y = Im [ F 1 ] , F z = Re [ F 2 ] ,
F 1 = 2 π ε k 2 | E 0 | 2 n , m [ c 11 F 1 ( 1 ) c 12 F 1 ( 2 ) + c 13 F 1 ( 3 ) ] , F 2 = 4 π ε k 2 | E 0 | 2 n , m [ c 21 F 2 ( 1 ) + c 22 F 2 ( 2 ) ] ,
c 11 = [ ( n m ) ( n + m + 1 ) n 2 ( n + 1 ) 2 ] 1 / 2 , c 12 [ n ( n + 2 ) ( n + m + 1 ) ( n + m + 2 ) ( n + 1 ) 2 ( 2 n + 1 ) ( 2 n + 3 ) ] 1 / 2 , c 13 = [ n ( n + 2 ) ( n m ) ( n m + 1 ) ( n + 1 ) 2 ( 2 n + 1 ) ( 2 n + 3 ) ] 1 / 2 , c 21 = [ n ( n + 2 ) ( n m + 1 ) ( n + m + 1 ) ( n + 1 ) 2 ( 2 n + 1 ) ( 2 n + 3 ) ] 1 / 2 , c 22 = m n ( n + 1 ) ,
F 1 ( 1 ) = a ˜ m n b ˜ m 1 n + b ˜ m n a ˜ m 1 n p ˜ m n q ˜ m 1 n q ˜ m n p ˜ m 1 n , F 1 ( 2 ) = a ˜ m n a ˜ m 1 n 1 + b ˜ m n b ˜ m 1 n 1 p ˜ m n p ˜ m 1 n 1 q ˜ m n q ˜ m 1 n 1 , F 1 ( 3 ) = a ˜ m n 1 a ˜ m 1 n + b ˜ m n 1 b ˜ m 1 n p ˜ m n 1 p ˜ m 1 n q ˜ m n 1 q ˜ m 1 n , F 2 ( 1 ) = a ˜ m n a ˜ m n 1 + b ˜ m n b ˜ m n 1 p ˜ m n p ˜ m n 1 q ˜ m n q ˜ m n 1 , F 2 ( 2 ) = a ˜ m n b ˜ m n * p ˜ m n q ˜ m n * ,
a ˜ m n = a m n 1 2 p m n , p ˜ m n = 1 2 p m n , b ˜ m n = b m n 1 2 q m n , q ˜ m n = 1 2 q m n .
p m n = i 1 n 4 π E 0 γ m n 1 / 2 k r j n ( k r ) θ = 0 π ϕ = 0 2 π [ e r E ( r , θ , ϕ ) ] P n m ( cos θ ) e i m ϕ sin θ d θ d ϕ , q m n = i n Z 4 π E 0 γ m n 1 / 2 k r j n ( k r ) θ = 0 π ϕ = 0 2 π [ e r H ( r , θ , ϕ ) ] P n m ( cos θ ) e i m ϕ sin θ d θ d ϕ ,
p m n = k 2 4 π 2 w x w y γ m n 1 / 2 e α x 3 + α y 3 3 α = 0 π / 2 d α sin α e i k z b cos α e k 2 w x 2 α x + w y 2 α y 2 sin 2 α × { 1 2 ( p x + i p y ) ( τ m n π m n cos α ) I c ( m + 1 ) +   1 2 ( p x i p y ) ( τ m n + π m n cos α ) I c ( m 1 ) , linear and circular polarization τ m n I c ( m ) , radial polarization i π m n cos α I c ( m ) , azimuthal polarization
q m n = k 2 4 π 2 w x w y γ m n 1 / 2 e α x 3 + α y 3 3 α = 0 π / 2 d α sin α e i k z b cos α e k 2 w x 2 α x + w y 2 α y 2 sin 2 α × { 1 2 ( p x + i p y ) ( π m n τ m n cos α ) I c ( m + 1 ) +   1 2 ( p x i p y ) ( π m n + τ m n cos α ) I c ( m 1 ) , linear and circular polarization π m n I c ( m ) , radial polarization i τ m n cos α I c ( m ) , azimuthal polarization
I c ( m ) = β = 0 2 π d β e i m β e i ( t 1 sin β + s 1 cos β + t 3 sin 3 β + s 3 cos 3 β ) + s 2 cos 2 β ,
E ( x , y , z ) = d k x d k y E 0 4 π 2 w x w y e 1 3 ( α x i k x w x ) 3 × e 1 3 ( α y i k y w y ) 3 e i ( k x k b + k y y b + k z z b ) e i ( k x x + k y y + k z z ) s ( α , β ) ,
H ( x , y , z ) = 1 i ω μ 0 μ b × E ( x , y , z ) .
p m n = C 0 ϕ = 0 2 π θ = 0 π α = 0 π / 2 β = 0 2 π d β d α d θ d ϕ C 1 C 2 e i m ϕ e i k r cos α cos θ e i k r sin α sin θ cos ( β ϕ ) × sin α sin θ P n m ( cos θ ) × [ cos θ cos α ( s x cos β + s y sin β ) + s x cos ϕ cos α sin θ + s y sin ϕ cos α sin θ ] ,
C 0 = i 1 n 4 π | E 0 | γ m n 1 / 2 k r j n ( k r ) , C 1 = E 0 4 π 2 w x w y k 2 e 1 3 ( α x i k w x cos β sin α ) 3 + 1 3 ( α y i k w y sin β sin α ) 3 , C 2 = e i k [ x b sin α cos β + y b sin α sin β + z b cos α ] .
q m n = C 0 ϕ = 0 2 π θ = 0 π α = 0 π / 2 β = 0 2 π d β d α d θ d ϕ C 1 C 2 e i m ϕ e i k r cos a cos θ e i k r sin α sin θ cos ( β ϕ ) × sin α sin θ P n m ( cos θ ) { sin α cos α cos θ ( s y cos β s x sin β ) cos ϕ sin θ [ s x sin 2 α sin β cos β + s y ( cos 2 α + sin 2 α sin 2 β ) ] + sin ϕ sin θ [ s x ( cos 2 α + sin 2 α cos 2 β + s y sin 2 α sin β cos β ) ] } ,
C 0 = i n Z 4 π | E 0 | γ m n 1 / 2 k r j n ( k r ) = i Z C 0 , C 1 = C 1 k ω μ 0 μ b = C 1 1 Z ,
0 2 π e i x cos ( β ϕ ) e i m ϕ [ cos ϕ sin ϕ 1 ] d ϕ = π i m e i m β [ i J m 1 ( x ) e i β i J m + 1 ( x ) e i β J m 1 ( x ) e i β + J m + 1 ( x ) e i β 2 J m ( x ) ] ,
J m ( x ) = 1 2 π 0 2 π exp ( i x sin ϕ i m ϕ ) d ϕ .
p m n = θ = 0 π α = 0 π / 2 β = 0 2 π d β d α d θ d C 0 C 1 C 2 e i k r cos α cos θ sin α sin θ P n m ( cos θ ) ( π ) i m e i m β × { cos θ cos α ( s x cos β + s y sin β ) ( 2 ) J m ( k r sin α sin θ ) + s x cos α sin θ [ i J m 1 ( k r sin α sin θ ) e i β i J m + 1 ( k r sin α sin θ ) e i β ] + s y cos α sin θ [ J m 1 ( k r sin α sin θ ) e i β + J m + 1 ( k r sin α sin θ ) e i β ] ,
q m n = θ = 0 π α = 0 π / 2 β = 0 2 π d β d α d θ d C 0 C 1 C 2 e i k r cos α cos θ sin α sin θ P n m ( cos θ ) ( π ) i m e i m β × { sin α cos α cos θ ( s y cos β s x sin β ) ( 2 ) J m ( k r sin α sin θ ) sin θ [ s x sin 2 α sin β cos β + s y ( cos 2 α + sin 2 α sin 2 β ) ] × [ i J m 1 ( k r sin α sin θ ) e i β i J m + 1 ( k r sin α sin θ ) e i β ] + sin θ [ s x ( cos 2 α + sin 2 α cos 2 β + s y sin 2 α sin β cos β ) ] × [ J m 1 ( k r sin α sin θ ) e i β + J m + 1 ( k r sin α sin θ ) e i β ] } .
J m 1 ( x ) = m x J m ( x ) + J m ( x ) , J m + 1 ( x ) = m x J m ( x ) J m ( x ) ,
J m ( k r sin α sin θ ) = d J m ( k r sin α sin θ ) d ( k r sin α sin θ ) = 1 k r sin θ cos α d J m ( k r sin α sin θ ) d α ,
d d α [ e i k r cos α cos θ P n m ( cos θ ) J m ( k r sin α sin θ ) ] = i k r sin α cos θ e i k r cos α cos θ P n m ( cos θ ) J m ( k r sin α sin θ ) + e i k r cos α cos θ P n m ( cos θ ) d m J m ( k r sin α sin θ ) d α .
d J m ( x ) d α = i k r sin α cos θ J m ( x ) + d C θ d α e i k r cos α cos θ / P n m ( cos θ ) .
p m n = α = 0 π / 2 β = 0 2 π θ = 0 π d β d α d θ ( π ) i m C 0 C 1 C 2 e i m β 1 k r × 2 sin θ [ d C θ d α i sin α ( s x cos β + s y sin β ) + C θ m cos α ( s y cos β s x sin β ) ] ,
q m n = α = 0 π / 2 β = 0 2 π θ = 0 π d β d α d θ ( π ) i m C 0 C 1 C 2 e i m β 1 k r × 2 sin θ [ d C θ d α i sin α cos α ( s x sin β s y cos β ) + C θ m ( s x cos β + s y sin β ) ] .
0 π C θ sin θ d θ = 2 i n m P n m ( cos α ) j n ( k r ) ,
d d α 0 π C θ sin θ d θ = 0 π d C θ d α sin θ d θ = 2 i n m j n ( k r ) d d α P n m ( cos α ) .
p m n = α = 0 π / 2 β = 0 2 π d β α C 0 C 1 C 2 ( π ) 4 i n e i m β j n ( k r ) k r × [ i d P n m ( cos α ) d α sin α ( s x cos β + s y sin β ) + m P n m ( cos α ) cos α ( s x sin β + s y cos β ) ] ,
q m n = α = 0 π / 2 β = 0 2 π d β d α C 0 C 1 C 2 ( π ) 4 i n e i m β j n ( k r ) k r × [ i d P n m ( cos α ) d α sin α ( s x sin β s y cos β ) + m P n m ( cos α ) cos α ( s x cos β + s y sin β ) ] .
π m n π m n ( cos α ) = m sin α P n m ( cos α ) , τ m n τ m n ( cos α ) = d d α P n m ( cos α ) ,
p m n = α = 0 π / 2 β = 0 2 π d β d α C 0 C 1 C 2 ( π ) 4 i n e i m β j n ( k r ) k r × sin α [ i ( s x cos β + s y sin β ) τ m n + ( s x sin β + s y cos β ) cos α π m n ] ,
q m n = α = 0 π / 2 β = 0 2 π d β d α C 0 C 1 C 2 ( π ) 4 i n e i m β j n ( k r ) k r × sin α [ i ( s x sin β s y cos β ) cos α τ m n + ( s x cos β + s y sin β ) π m n ] .
p m n = k 2 4 π 2 w x w y γ m n 1 / 2 α = 0 π / 2 β = 0 2 π d β d α × sin α [ ( s x cos β + s y sin β ) τ m n + i ( s x sin β s y cos β ) cos α π m n ] × e i m β e 1 3 ( α x i k w x cos β sin α ) 3 + 1 3 ( α y i k w y sin β sin α ) 3 e i k [ x b sin α cos β + y b sin α sin β + z b cos α ] ,
q m n = k 2 4 π 2 w x w y γ m n 1 / 2 α = 0 π / 2 β = 0 2 π d β d α × sin α [ ( s x cos β + s y sin β ) π m n + i ( s x sin β s y cos β ) cos α τ m n ] × e i m β e 1 3 ( α x i k w x cos β sin α ) 3 + 1 3 ( α y i k w y sin β sin α ) 3 e i k [ x b sin α cos β + y b sin α sin β + z b cos α ] .
cos 2 β = 1 + cos 2 β 2 , sin 2 β = 1 sin 2 β 2 , cos 3 β = 3 cos β + cos 3 β 4 , sin 3 β = 3 sin β sin 3 β 4 ,
p m n = k 2 4 π 2 w x w y γ m n 1 / 2 α = 0 π / 2 β = 0 2 π d β d α × sin α [ ( s x cos β + s y sin β ) τ m n + i ( s x sin β s y cos β ) cos α π m n ] × e i k z b cos α e t 0 e i m β e i t 1 sin β e i s 1 cos β e s 2 cos 2 β e i t 3 sin 3 β e i s 3 cos 3 β ,
q m n = k 2 4 π 2 w x w y γ m n 1 / 2 α = 0 π / 2 β = 0 2 π d β d α × sin α [ ( s x cos β + s y sin β ) π m n + i ( s x sin β s y cos β ) cos α τ m n ] × e i k z b cos α e t 0 e i m β e i t 1 sin β e i s 1 cos β e s 2 cos 2 β e i t 3 sin 3 β e i s 3 cos 3 β ,

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