Abstract

Based on an expansion formula for unit dyadic in terms of the vector spherical wave functions, we derive explicit partial wave coefficients for a complex wave vector field that is characterized by a single wave vector with three Cartesian components being arbitrarily constant complex except subject to lossless background constraint and thus includes evanescent waves and simple plasmonic fields as its two special cases. A recurrence method is then proposed to evaluate the partial wave expansion coefficients numerically up to arbitrary order of expansion, offering an efficient tool for the scattering of generic electromagnetic fields that can be modelled by a superposition of the complex wave vector fields such as the evanescent and plasmonic waves. Our approach is validated by analytically working out the integration in the conventional, more cumbersome, projection approach. Comparison of optical forces on a particle in evanescent and plasmonic fields with previous results shows perfect agreement, thereby further corroborating our approach. As examples of its application, we calculate optical force and torque exerting on particles residing in a plasmonic field, with large particle size where the conventional projection method based on the direct numerical integration is unadapted due to the difficulty in convergence. It is found that the direction of optical torque stays parallel to the direction of spin of optical field for some field polarizations and changes for some other polarizations, as the particle radius R varies.

© 2017 Optical Society of America

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References

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    [Crossref] [PubMed]
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2016 (1)

2015 (2)

G. Gouesbet and J. A. Lock, “On the description of electromagnetic arbitrary shaped beams: The relationship between beam shape coefficients and plane wave spectra,” J. Quant. Spectrosc. Radiat. Transfer 162, 18–30 (2015).
[Crossref]

J. A. Lock, “Scattering of the evanescent components in the angular spectrum of a tightly focused electromagnetic beam by a spherical particle,” J. Quant. Spectrosc. Radiat. Transfer 162, 95–102 (2015).
[Crossref]

2014 (3)

A. C. Durand and C. Genet, “Transverse spinning of a sphere in a plasmonic field,” Phys. Rev. A 89(3), 033841 (2014).
[Crossref]

M. I. Mishchenko, N. T. Zakharov, N. G. Khlebtsov, T. Wriedt, and G. Videen, “Comprehensive thematic T-matrix reference database: A 2013–2014 update,” J. Quant. Spectrosc. Radiat. Transfer 146, 349–354 (2014), and references therein.
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
[Crossref] [PubMed]

2013 (4)

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Mie scattering and optical forces from evanescent fields: A complex-angle approach,” Opt. Express 21(6), 7082–7095 (2013).
[Crossref] [PubMed]

E. Almaas and I. Brevik, “Possible sorting mechanism for microparticles in an evanescent field,” Phys. Rev. A 87(6), 063826 (2013).
[Crossref]

N. Wang, J. Chen, S. Y. Liu, and Z. F. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87(6), 063812 (2013).
[Crossref]

A. Cuche, A. C. Durand, E. Devaux, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Sorting nanoparticles with intertwined plasmonic and thermo-hydrodynamical forces,” Nano Lett. 13, 4230–4235 (2013).
[Crossref] [PubMed]

2012 (1)

2010 (1)

2009 (2)

J. M. Taylor and G. D. Love, “Multipole expansion of Bessel and Gaussian beams for Mie scattering calculations,” J. Opt. Soc. Am. A 26(2), 278–282 (2009).
[Crossref]

E. Xifré-Pérez, F. J. García de Abajo, R. Fenollosa, and F. Meseguer, “Photonic binding in silicon-colloid micro-cavities,” Phys. Rev. Lett. 103(10), 103902 (2009).
[Crossref]

2007 (2)

S. Koumandos, “On a class of integrals involving a Bessel function times Gegenbauer polynomials,” Internat. J. Math. Math. Sci. 2007, 1–5 (2007).
[Crossref]

P. J. Cregg and P. Svedlindh, “Comment on ‘Analytical results for a Bessel function times Legendre polynomials class integrals’,” J. Phys. A: Math. Theor. 40, 14029 (2007).
[Crossref]

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(16), 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(18), L293–L296 (2006).
[Crossref]

2005 (3)

J. Ng, C. T. Chan, P. Sheng, and Z. F. Lin, “Strong optical force induced by morphology-dependent resonances,” Opt. Lett. 30(15), 1956–1958 (2005).
[Crossref] [PubMed]

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005).
[Crossref]

M. K. Liu, N. Ji, Z. F. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E 72(5), 056610 (2005).
[Crossref]

2004 (1)

Z. F. Lin and S. T. Chui, “Scattering by optically anisotropic magnetic particle,” Phys. Rev. E 69(5), 056614 (2004).
[Crossref]

1999 (1)

Y. G. Song, S. Chang, and J. H. Jo, “Optically induced rotation of combined Mie particles within an evanescent field of a Gaussian beam,” Jpn. J. Appl. Phys. 38, L380–L383 (1999).
[Crossref]

1998 (2)

S. Chang and S. S. Lee, “Optical torque exerted on a sphere in the evanescent field of a circularly-polarized Gaussian laser beam,” Opt. Commun. 151(4–6), 286–296 (1998).
[Crossref]

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152(4–6), 376–384 (1998).
[Crossref]

1997 (1)

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56(1) 1102–1112 (1997).
[Crossref]

1996 (1)

Ø. Farsund and B. U. Felderhof, “Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnatic radiation field,” Physica A 227(1–2), 108–130 (1996).
[Crossref]

1995 (2)

1994 (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)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[Crossref]

1980 (1)

1971 (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3(4), 825–839 (1971).
[Crossref]

1967 (1)

G. S. Joyce, “Classical Heisenberg model,” Phys. Rev. 155(2), 478–491 (1967).
[Crossref]

1929 (1)

B. Podolsky and L. Pauling, “The momentum distribution in hydrogen-like atoms,” Phys. Rev. 34(1), 109–116 (1929).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1972).

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]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[Crossref]

Almaas, E.

E. Almaas and I. Brevik, “Possible sorting mechanism for microparticles in an evanescent field,” Phys. Rev. A 87(6), 063826 (2013).
[Crossref]

E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995).
[Crossref]

Arfken, G. B.

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 6th ed. (Elsevier Academic, 2005).

Barbosa, L. C.

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(18), 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(16), 2477–2479 (2006).
[Crossref] [PubMed]

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]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[Crossref]

Bekshaev, A. Y.

Bliokh, K. Y.

Bohren, C. F.

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

Brevik, I.

E. Almaas and I. Brevik, “Possible sorting mechanism for microparticles in an evanescent field,” Phys. Rev. A 87(6), 063826 (2013).
[Crossref]

E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995).
[Crossref]

Cesar, C. L.

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(16), 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(18), L293–L296 (2006).
[Crossref]

Chan, C. T.

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005).
[Crossref]

J. Ng, C. T. Chan, P. Sheng, and Z. F. Lin, “Strong optical force induced by morphology-dependent resonances,” Opt. Lett. 30(15), 1956–1958 (2005).
[Crossref] [PubMed]

Chang, S.

Y. G. Song, S. Chang, and J. H. Jo, “Optically induced rotation of combined Mie particles within an evanescent field of a Gaussian beam,” Jpn. J. Appl. Phys. 38, L380–L383 (1999).
[Crossref]

S. Chang and S. S. Lee, “Optical torque exerted on a sphere in the evanescent field of a circularly-polarized Gaussian laser beam,” Opt. Commun. 151(4–6), 286–296 (1998).
[Crossref]

Chen, J.

N. Wang, J. Chen, S. Y. Liu, and Z. F. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87(6), 063812 (2013).
[Crossref]

J. Chen, J. Ng, P. Wang, and Z. F. Lin, “Analytical partial wave expansion of vector Bessel beam and its application to optical binding”, Opt. Lett. 35(10), 1674–1676 (2010).
[Crossref] [PubMed]

Christodoulides, D. N.

Chui, S. T.

M. K. Liu, N. Ji, Z. F. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E 72(5), 056610 (2005).
[Crossref]

Z. F. Lin and S. T. Chui, “Scattering by optically anisotropic magnetic particle,” Phys. Rev. E 69(5), 056614 (2004).
[Crossref]

Cregg, P. J.

P. J. Cregg and P. Svedlindh, “Comment on ‘Analytical results for a Bessel function times Legendre polynomials class integrals’,” J. Phys. A: Math. Theor. 40, 14029 (2007).
[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(16), 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(18), L293–L296 (2006).
[Crossref]

Cuche, A.

A. Cuche, A. C. Durand, E. Devaux, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Sorting nanoparticles with intertwined plasmonic and thermo-hydrodynamical forces,” Nano Lett. 13, 4230–4235 (2013).
[Crossref] [PubMed]

Devaux, E.

A. Cuche, A. C. Durand, E. Devaux, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Sorting nanoparticles with intertwined plasmonic and thermo-hydrodynamical forces,” Nano Lett. 13, 4230–4235 (2013).
[Crossref] [PubMed]

Doicu, A.

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152(4–6), 376–384 (1998).
[Crossref]

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
[Crossref]

Durand, A. C.

A. C. Durand and C. Genet, “Transverse spinning of a sphere in a plasmonic field,” Phys. Rev. A 89(3), 033841 (2014).
[Crossref]

A. Cuche, A. C. Durand, E. Devaux, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Sorting nanoparticles with intertwined plasmonic and thermo-hydrodynamical forces,” Nano Lett. 13, 4230–4235 (2013).
[Crossref] [PubMed]

Ebbesen, T. W.

A. Cuche, A. C. Durand, E. Devaux, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Sorting nanoparticles with intertwined plasmonic and thermo-hydrodynamical forces,” Nano Lett. 13, 4230–4235 (2013).
[Crossref] [PubMed]

Eremin, Y. A.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
[Crossref]

Fardad, S.

Farsund, Ø.

Ø. Farsund and B. U. Felderhof, “Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnatic radiation field,” Physica A 227(1–2), 108–130 (1996).
[Crossref]

Felderhof, B. U.

Ø. Farsund and B. U. Felderhof, “Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnatic radiation field,” Physica A 227(1–2), 108–130 (1996).
[Crossref]

Fenollosa, R.

E. Xifré-Pérez, F. J. García de Abajo, R. Fenollosa, and F. Meseguer, “Photonic binding in silicon-colloid micro-cavities,” Phys. Rev. Lett. 103(10), 103902 (2009).
[Crossref]

Fontes, A.

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(16), 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(18), L293–L296 (2006).
[Crossref]

García de Abajo, F. J.

E. Xifré-Pérez, F. J. García de Abajo, R. Fenollosa, and F. Meseguer, “Photonic binding in silicon-colloid micro-cavities,” Phys. Rev. Lett. 103(10), 103902 (2009).
[Crossref]

Genet, C.

A. C. Durand and C. Genet, “Transverse spinning of a sphere in a plasmonic field,” Phys. Rev. A 89(3), 033841 (2014).
[Crossref]

A. Cuche, A. C. Durand, E. Devaux, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Sorting nanoparticles with intertwined plasmonic and thermo-hydrodynamical forces,” Nano Lett. 13, 4230–4235 (2013).
[Crossref] [PubMed]

Gouesbet, G.

G. Gouesbet and J. A. Lock, “On the description of electromagnetic arbitrary shaped beams: The relationship between beam shape coefficients and plane wave spectra,” J. Quant. Spectrosc. Radiat. Transfer 162, 18–30 (2015).
[Crossref]

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, 2011).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Elsevier Academic, 2007).

Gréhan, G.

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, 2011).
[Crossref]

Halas, N. J.

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56(1) 1102–1112 (1997).
[Crossref]

Harris, F. E.

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 6th ed. (Elsevier Academic, 2005).

Huffman, D. R.

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

Hutchison, J. A.

A. Cuche, A. C. Durand, E. Devaux, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Sorting nanoparticles with intertwined plasmonic and thermo-hydrodynamical forces,” Nano Lett. 13, 4230–4235 (2013).
[Crossref] [PubMed]

Jackson, J. D.

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

Jacob, Z.

Ji, N.

M. K. Liu, N. Ji, Z. F. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E 72(5), 056610 (2005).
[Crossref]

Jo, J. H.

Y. G. Song, S. Chang, and J. H. Jo, “Optically induced rotation of combined Mie particles within an evanescent field of a Gaussian beam,” Jpn. J. Appl. Phys. 38, L380–L383 (1999).
[Crossref]

Joyce, G. S.

G. S. Joyce, “Classical Heisenberg model,” Phys. Rev. 155(2), 478–491 (1967).
[Crossref]

Khlebtsov, N. G.

M. I. Mishchenko, N. T. Zakharov, N. G. Khlebtsov, T. Wriedt, and G. Videen, “Comprehensive thematic T-matrix reference database: A 2013–2014 update,” J. Quant. Spectrosc. Radiat. Transfer 146, 349–354 (2014), and references therein.
[Crossref]

Kong, J. A.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (John Wiley & Sons, 1985).

Koumandos, S.

S. Koumandos, “On a class of integrals involving a Bessel function times Gegenbauer polynomials,” Internat. J. Math. Math. Sci. 2007, 1–5 (2007).
[Crossref]

Lee, S. S.

S. Chang and S. S. Lee, “Optical torque exerted on a sphere in the evanescent field of a circularly-polarized Gaussian laser beam,” Opt. Commun. 151(4–6), 286–296 (1998).
[Crossref]

Lin, Z. F.

N. Wang, J. Chen, S. Y. Liu, and Z. F. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87(6), 063812 (2013).
[Crossref]

J. Chen, J. Ng, P. Wang, and Z. F. Lin, “Analytical partial wave expansion of vector Bessel beam and its application to optical binding”, Opt. Lett. 35(10), 1674–1676 (2010).
[Crossref] [PubMed]

J. Ng, C. T. Chan, P. Sheng, and Z. F. Lin, “Strong optical force induced by morphology-dependent resonances,” Opt. Lett. 30(15), 1956–1958 (2005).
[Crossref] [PubMed]

M. K. Liu, N. Ji, Z. F. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E 72(5), 056610 (2005).
[Crossref]

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005).
[Crossref]

Z. F. Lin and S. T. Chui, “Scattering by optically anisotropic magnetic particle,” Phys. Rev. E 69(5), 056614 (2004).
[Crossref]

Liu, M. K.

M. K. Liu, N. Ji, Z. F. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E 72(5), 056610 (2005).
[Crossref]

Liu, S. Y.

N. Wang, J. Chen, S. Y. Liu, and Z. F. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87(6), 063812 (2013).
[Crossref]

Lock, J. A.

G. Gouesbet and J. A. Lock, “On the description of electromagnetic arbitrary shaped beams: The relationship between beam shape coefficients and plane wave spectra,” J. Quant. Spectrosc. Radiat. Transfer 162, 18–30 (2015).
[Crossref]

J. A. Lock, “Scattering of the evanescent components in the angular spectrum of a tightly focused electromagnetic beam by a spherical particle,” J. Quant. Spectrosc. Radiat. Transfer 162, 95–102 (2015).
[Crossref]

Love, G. D.

Mackowski, D. W.

Mechelen, T. V.

Meseguer, F.

E. Xifré-Pérez, F. J. García de Abajo, R. Fenollosa, and F. Meseguer, “Photonic binding in silicon-colloid micro-cavities,” Phys. Rev. Lett. 103(10), 103902 (2009).
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko, N. T. Zakharov, N. G. Khlebtsov, T. Wriedt, and G. Videen, “Comprehensive thematic T-matrix reference database: A 2013–2014 update,” J. Quant. Spectrosc. Radiat. Transfer 146, 349–354 (2014), and references therein.
[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(18), 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(16), 2477–2479 (2006).
[Crossref] [PubMed]

Ng, J.

Nori, F.

Padilha, L. A.

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(16), 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(18), L293–L296 (2006).
[Crossref]

Pauling, L.

B. Podolsky and L. Pauling, “The momentum distribution in hydrogen-like atoms,” Phys. Rev. 34(1), 109–116 (1929).
[Crossref]

Podolsky, B.

B. Podolsky and L. Pauling, “The momentum distribution in hydrogen-like atoms,” Phys. Rev. 34(1), 109–116 (1929).
[Crossref]

Raether, H.

H. Raether, Surface Plasmons (Springer, 1986).

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(18), 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(16), 2477–2479 (2006).
[Crossref] [PubMed]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Elsevier Academic, 2007).

Salandrino, A.

Sarkar, D.

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56(1) 1102–1112 (1997).
[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]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[Crossref]

Sheng, P.

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005).
[Crossref]

J. Ng, C. T. Chan, P. Sheng, and Z. F. Lin, “Strong optical force induced by morphology-dependent resonances,” Opt. Lett. 30(15), 1956–1958 (2005).
[Crossref] [PubMed]

Shin, R. T.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (John Wiley & Sons, 1985).

Song, Y. G.

Y. G. Song, S. Chang, and J. H. Jo, “Optically induced rotation of combined Mie particles within an evanescent field of a Gaussian beam,” Jpn. J. Appl. Phys. 38, L380–L383 (1999).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1972).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Svedlindh, P.

P. J. Cregg and P. Svedlindh, “Comment on ‘Analytical results for a Bessel function times Legendre polynomials class integrals’,” J. Phys. A: Math. Theor. 40, 14029 (2007).
[Crossref]

Taylor, J. M.

Tsang, L.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (John Wiley & Sons, 1985).

Videen, G.

M. I. Mishchenko, N. T. Zakharov, N. G. Khlebtsov, T. Wriedt, and G. Videen, “Comprehensive thematic T-matrix reference database: A 2013–2014 update,” J. Quant. Spectrosc. Radiat. Transfer 146, 349–354 (2014), and references therein.
[Crossref]

Wang, N.

N. Wang, J. Chen, S. Y. Liu, and Z. F. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87(6), 063812 (2013).
[Crossref]

Wang, P.

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3(4), 825–839 (1971).
[Crossref]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, 1944).

Weber, H. J.

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 6th ed. (Elsevier Academic, 2005).

Wiscombe, W. J.

Wriedt, T.

M. I. Mishchenko, N. T. Zakharov, N. G. Khlebtsov, T. Wriedt, and G. Videen, “Comprehensive thematic T-matrix reference database: A 2013–2014 update,” J. Quant. Spectrosc. Radiat. Transfer 146, 349–354 (2014), and references therein.
[Crossref]

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152(4–6), 376–384 (1998).
[Crossref]

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
[Crossref]

Xifré-Pérez, E.

E. Xifré-Pérez, F. J. García de Abajo, R. Fenollosa, and F. Meseguer, “Photonic binding in silicon-colloid micro-cavities,” Phys. Rev. Lett. 103(10), 103902 (2009).
[Crossref]

Xu, Y. L.

Zakharov, N. T.

M. I. Mishchenko, N. T. Zakharov, N. G. Khlebtsov, T. Wriedt, and G. Videen, “Comprehensive thematic T-matrix reference database: A 2013–2014 update,” J. Quant. Spectrosc. Radiat. Transfer 146, 349–354 (2014), and references therein.
[Crossref]

Appl. Opt. (2)

Internat. J. Math. Math. Sci. (1)

S. Koumandos, “On a class of integrals involving a Bessel function times Gegenbauer polynomials,” Internat. J. Math. Math. Sci. 2007, 1–5 (2007).
[Crossref]

J. Appl. Phys. (2)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[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(10), 4594–4602 (1989).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

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(18), L293–L296 (2006).
[Crossref]

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

P. J. Cregg and P. Svedlindh, “Comment on ‘Analytical results for a Bessel function times Legendre polynomials class integrals’,” J. Phys. A: Math. Theor. 40, 14029 (2007).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (3)

G. Gouesbet and J. A. Lock, “On the description of electromagnetic arbitrary shaped beams: The relationship between beam shape coefficients and plane wave spectra,” J. Quant. Spectrosc. Radiat. Transfer 162, 18–30 (2015).
[Crossref]

J. A. Lock, “Scattering of the evanescent components in the angular spectrum of a tightly focused electromagnetic beam by a spherical particle,” J. Quant. Spectrosc. Radiat. Transfer 162, 95–102 (2015).
[Crossref]

M. I. Mishchenko, N. T. Zakharov, N. G. Khlebtsov, T. Wriedt, and G. Videen, “Comprehensive thematic T-matrix reference database: A 2013–2014 update,” J. Quant. Spectrosc. Radiat. Transfer 146, 349–354 (2014), and references therein.
[Crossref]

Jpn. J. Appl. Phys. (1)

Y. G. Song, S. Chang, and J. H. Jo, “Optically induced rotation of combined Mie particles within an evanescent field of a Gaussian beam,” Jpn. J. Appl. Phys. 38, L380–L383 (1999).
[Crossref]

Nano Lett. (1)

A. Cuche, A. C. Durand, E. Devaux, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Sorting nanoparticles with intertwined plasmonic and thermo-hydrodynamical forces,” Nano Lett. 13, 4230–4235 (2013).
[Crossref] [PubMed]

Nat. Commun. (1)

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
[Crossref] [PubMed]

Opt. Commun. (2)

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152(4–6), 376–384 (1998).
[Crossref]

S. Chang and S. S. Lee, “Optical torque exerted on a sphere in the evanescent field of a circularly-polarized Gaussian laser beam,” Opt. Commun. 151(4–6), 286–296 (1998).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Optica (1)

Phys. Rev. (2)

B. Podolsky and L. Pauling, “The momentum distribution in hydrogen-like atoms,” Phys. Rev. 34(1), 109–116 (1929).
[Crossref]

G. S. Joyce, “Classical Heisenberg model,” Phys. Rev. 155(2), 478–491 (1967).
[Crossref]

Phys. Rev. A (3)

N. Wang, J. Chen, S. Y. Liu, and Z. F. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87(6), 063812 (2013).
[Crossref]

A. C. Durand and C. Genet, “Transverse spinning of a sphere in a plasmonic field,” Phys. Rev. A 89(3), 033841 (2014).
[Crossref]

E. Almaas and I. Brevik, “Possible sorting mechanism for microparticles in an evanescent field,” Phys. Rev. A 87(6), 063826 (2013).
[Crossref]

Phys. Rev. B (1)

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005).
[Crossref]

Phys. Rev. D (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3(4), 825–839 (1971).
[Crossref]

Phys. Rev. E (3)

Z. F. Lin and S. T. Chui, “Scattering by optically anisotropic magnetic particle,” Phys. Rev. E 69(5), 056614 (2004).
[Crossref]

M. K. Liu, N. Ji, Z. F. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E 72(5), 056610 (2005).
[Crossref]

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56(1) 1102–1112 (1997).
[Crossref]

Phys. Rev. Lett. (1)

E. Xifré-Pérez, F. J. García de Abajo, R. Fenollosa, and F. Meseguer, “Photonic binding in silicon-colloid micro-cavities,” Phys. Rev. Lett. 103(10), 103902 (2009).
[Crossref]

Physica A (1)

Ø. Farsund and B. U. Felderhof, “Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnatic radiation field,” Physica A 227(1–2), 108–130 (1996).
[Crossref]

Other (12)

H. Raether, Surface Plasmons (Springer, 1986).

See, http://functions.wolfram.com/PDF/LegendreP2General.pdf . It is noted that Eq. (9) in this paper differs from that given in http://functions.wolfram.com/07.08.26.0005.01 by a factor of (−1)m.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1972).

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

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, 2011).
[Crossref]

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
[Crossref]

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (John Wiley & Sons, 1985).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

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

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 6th ed. (Elsevier Academic, 2005).

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, 1944).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Elsevier Academic, 2007).

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Figures (2)

Fig. 1
Fig. 1 Normalized optical force components fx, fz and torque T y acting on a gold sphere in a plasmonic field versus sphere radius R. (a) The force components fx (left axis, black line) and fz (right axis, blue line) as a function of R. The red circles and squares denote fx and fz from Ref. [14], while the continuous curves represent results based on our approach; (b) The same as panel (a) except for optical torque T y . The open circles and continuous curve exhibit, respectively, results from Ref. [14] and the calculation based on our approach; (c) The force components fx (left axis, black line) and fz (right axis, blue line) evaluated based on our approach for larger range of R including the radius regime (R > 2μm) where numerical integration for PWECs is difficult to converge; (d) The same as panel (c) except for optical torque T y .
Fig. 2
Fig. 2 The direction cosines of optical torque ( T x / T t , T y / T t , T z / T t ), as well as the magnitude of torque, as a function of particle radius R when the a dissipative gold particle is immersed in a complex-k field characterized by the same wave vector components kx/k0 = −0.40 + 0.30i, ky/k0 = 0.90 − 0.20i, and kz/k0 determined by Eq. (3) with (a) Ex = −0.68−0.10i, Ey = 0.45+0.22i, with Ez calculated by Eq. (4); (b) Ex = 0.87 − 0.36i, Ey = 0.19 − 0.53i, with Ez calculated by Eq. (4) as well. The direction of torque may keep fixed to the spin direction Eq. (24) of the optical field, or change as the R increases, dependent on the polarization of the complex-k field.

Equations (80)

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

E inc = E e i k r = E 0 ( E x e x + E y e y + E z e z ) e i ( k x r x + k y r y + k z r z ) ,
r = r x e x + r y e y + r z e z , k = k x e x + k y e y + k z e z ,
k k = k x 2 + k y 2 + k z 2 = k 2 = ω 2 ε μ > 0 ,
k x E x + k y E y + k z E z = 0 .
E inc = i E 0 n = 1 m = n n E m n [ p m n N m n ( k , r ) + q m n M m n ( k , r ) ] ,
E m n = ( 2 n + 1 ) i n ( n m ) ! ( n + m ) ! .
L m n ( k , r ) = [ τ m n ( cos θ ) e θ + i τ m n ( cos θ ) e ϕ ] j n ( k r ) k r e i m ϕ + e r P n m ( cos θ ) 1 k d [ j n ( k r ) ] d r e i m ϕ M m n ( k , r ) = [ i τ m n ( cos θ ) e θ τ m n ( cos θ ) e ϕ ] j n ( k r ) e i m ϕ , N m n ( k , r ) = [ τ m n ( cos θ ) e θ + i τ m n ( cos θ ) e ϕ ] 1 k r d [ r j n ( k r ) ] d r e i m ϕ + e r n ( n + 1 ) P n m ( cos θ ) j n ( k r ) k r e i m ϕ ,
π m n ( z ) = m 1 z 2 P n m ( z ) , τ m n ( z ) = 1 z 2 d P n m ( z ) d z
P n m ( z ) = ( 1 ) m Γ ( 1 m ) ( 1 + z ) m / 2 ( 1 z ) m / 2   2 F 1 ( n , n + 1 ; 1 m ; 1 z 2 ) ,
P n m ( x ) = ( 1 x 2 ) m / 2 2 n n ! d n + m d x n + m ( x 2 1 ) n .
E = E 0 ( E x e x + E y e y + E z e z ) = E 0 ( p 1 θ ^ k + p 2 ϕ ^ k ) ,
k ^ = sin α cos β e x + sin α sin β e y + cos α e z , θ ^ k = cos α cos β e x + cos α sin β e y sin α e z , ϕ ^ k = sin β e x + cos β e y ,
k ^ θ ^ k = θ ^ k ϕ ^ k = k ^ ϕ ^ k = 0 , k ^ k ^ = θ ^ k θ ^ k = ϕ ^ k ϕ ^ k = 1 .
cos α = k z k , sin α = k ρ k , cos β = k x k ρ , sin β = k y k ρ ,
p 1 = k E z k ρ , p 2 = k x E y k y E x k ρ .
I ^ e i k r = n = 0 + m = n + n [ A m n ( 0 ) L m n ( k , r ) + B m n ( 0 ) M m n ( k , r ) + C m n ( 0 ) N m n ( k , r ) ] ,
A m n ( 0 ) = i E m n P n m ( cos α ) k ^ e i m β ,
B m n ( 0 ) = i E m n n ( n + 1 ) [ π m n ( cos α ) θ ^ k i τ m n ( cos α ) ϕ ^ k ] e i m β ,
C m n ( 0 ) = i E m n n ( n + 1 ) [ τ m n ( cos α ) θ ^ k i π m n ( cos α ) ϕ ^ k ] e i m β .
E inc = E e i k r = E I ^ e i k r = E 0 ( p 1 θ ^ k + p 2 ϕ ^ k ) n = 0 + m = n + n [ A m n ( 0 ) L m n ( k , r ) + B m n ( 0 ) M m n ( k , r ) + C m n ( 0 ) N m n ( k , r ) ] ,
p m n = 1 n ( n + 1 ) [ p 1 τ m n ( cos α ) i p 2 π m n ( cos α ) ] ( cos β i sin β ) m , q m n = 1 n ( n + 1 ) [ p 1 τ m n ( cos α ) i p 2 π m n ( cos α ) ] ( cos β i sin β ) m ,
π ˜ m , n = c m n π m n , τ ˜ m , n = c m n τ m n ,
c m n = ( 2 n + 1 ) ( n m ) ! n ( n + 1 ) ( n + m ) ! .
π ˜ m , n = ( 1 ) m π ˜ m , n , τ ˜ m , n = ( 1 ) m τ ˜ m , n ,
π ˜ n , n = [ n ( 2 n + 1 ) ( 1 z 2 ) 2 ( n + 1 ) ( n 1 ) ] 1 2 π ˜ n 1 , n 1 , π ˜ n 1 , n = [ ( n 1 ) ( 2 n + 1 ) ( n + 1 ) ] 1 2 z π ˜ n 1 , n 1 ,
π ˜ m , n = [ ( 2 n + 1 ) ( n 1 ) ( n + m 1 ) ( n m 1 ) ( n 2 ) ( n + 1 ) ( n m ) ( n + m ) n ( 2 n 3 ) ] 1 2 π ˜ m , n 2 + [ ( 2 n + 1 ) ( n 1 ) ( 2 n 1 ) ( n + 1 ) ( n m ) ( n + m ) ] 1 2 z π ˜ m , n 1 , 0 < m n 2 ,
τ ˜ 0 , n = [ ( 2 n + 1 ) ( 2 n 1 ) ( n 1 ) ( n + 1 ) ] 1 2 z τ ˜ 0 , n 1 [ ( 2 n + 1 ) n ( n 2 ) ( n 1 ) ( n + 1 ) ( 2 n 3 ) ] 1 2 τ ˜ 0 , n 2 , n > 2 ,
τ ˜ m , n = n z m π ˜ m , n 1 m [ ( n + m ) ( n m ) ( 2 n + 1 ) ( n 1 ) ( 2 n 1 ) ( n + 1 ) ] 1 2 π ˜ m , n 1 , m > 0 .
π ˜ 0 , n ( z ) = 0 , π ˜ 1 , 1 ( z ) = 3 2 , π ˜ 1 , 2 ( z ) = 5 2 z , π ˜ 2 , 2 ( z ) = 5 2 1 z 2 , τ ˜ 0 , 1 ( z ) = 6 2 1 z 2 , τ ˜ 1 , 1 ( z ) = 3 2 z , τ ˜ 0 , 2 ( z ) = 30 2 z 1 z 2 , τ ˜ 1 , 2 ( z ) = 5 2 ( 2 z 2 1 ) , τ ˜ 2 , 2 = 5 2 z 1 z 2 .  
k x k = ε m ε d + ε m , k y k = 0 , k z k = ε d ε d + ε m ,
n c = x + 4.05 x 1 / 3 + 2 ,
s = κ × η / | κ × η | ,
I ^ e i k r = n = 0 + m = n + n [ A m n ( 0 ) L m n ( k , r ) + B m n ( 0 ) M m n ( k , r ) + C m n ( 0 ) N m n ( k , r ) ] ,
V r N m n ( k , r ) N u v ( p , r ) d 3 r = 2 i n n ( n + 1 ) E m n π 2 k 2 δ ( k p ) δ m u δ n v ,
V r M m n ( k , r ) M u v ( p , r ) d 3 r = 2 i n n ( n + 1 ) E m n π 2 k 2 δ ( k p ) δ m u δ n v ,
V r L m n ( k , r ) L u v ( p , r ) d 3 r = 2 i n E m n π 2 k 2 δ ( k p ) δ m u δ n v ,
V r M m n ( k , r ) N u v ( p , r ) d 3 r = V r M m n ( k , r ) L u v ( p , r ) d 3 r = V r M m n ( k , r ) L u v ( p , r ) d 3 r = 0 ,
V r [ 0 L m n * ( p , r ) p 2 d p ] e i k r d 3 r = 2 i n π 2 E m n A m n ( 0 ) ,
V r [ 0 M m n * ( p , r ) p 2 d p ] e i k r d 3 r = 2 i n π 2 n ( n + 1 ) E m n B m n ( 0 ) ,
V r [ 0 N m n * ( p , r ) p 2 d p ] e i k r d 3 r = 2 i n π 2 n ( n + 1 ) E m n C m n ( 0 ) .
Y n m ( θ , ϕ ) = P n m ( cos θ ) e i m ϕ , Y n m ( θ , ϕ ) = ( 1 ) m ( n m ) ! ( n + m ) ! Y n m ,
P m n ( θ , ϕ ) = Y n m ( θ , ϕ ) e r = P n m ( cos θ ) e i m ϕ e r ,
B m n ( θ , ϕ ) = × [ Y n m ( θ , ϕ ) r ] = [ i π m n ( cos θ ) e θ τ m n ( cos θ ) e ϕ ] e i m ϕ ,
C m n ( θ , ϕ ) = r [ Y n m ( θ , ϕ ) ] = [ τ m n ( cos θ ) e θ i π m n ( cos θ ) e ϕ ] e i m ϕ .
e i k r = n = 0 m = n n i n ( 2 n + 1 ) j n ( k r ) ( 1 ) m Y n m ( θ , ϕ ) Y n m ( θ , ϕ ) ,
j n ( k r ) Y n m ( θ , ϕ ) = i n 4 π 4 π e i k r Y n m ( θ , ϕ ) d Ω ,
L m n ( k , r ) = 1 k [ j n ( k r ) Y n m ( θ , ϕ ) ] = ( i ) n 1 4 π 4 π e i k r P m n ( θ , ϕ ) d Ω ,
L m n ( k , r ) = i n 1 ( 1 ) m 4 π ( n + m ) ! ( n m ) ! 4 π e i k r k ^ Y n m ( θ , ϕ ) d Ω ,
0 L m n ( k , r ) k 2 d k = i n 1 ( 1 ) m 4 π ( n + m ) ! ( n m ) ! V r e i k ^ r k ^ Y n m ( θ , ϕ ) d 3 k ,
V r [ 0 L m n ( p , r ) p 2 d p ] e i k r d 3 r = 2 π 2 i n 1 P n m ( cos θ ) e i m ϕ k ^ .
A m n ( 0 ) = i E m n P n m ( cos θ ) e i m ϕ k ^ ,
M m n ( k , r ) = k r × L m n ( k , r ) = i n 4 π 4 π { k × [ e i k r Y n m ( θ , ϕ ) ] e i k r k × [ Y n m ( θ , ϕ ) ] } d Ω ,
M m n ( k , r ) = i n 4 π 4 π e i k r B m n ( θ , ϕ ) d Ω ,
0 M m n ( k , r ) k 2 d k = i n 4 π ( 1 ) m ( n + m ) ! ( n m ) ! V k e i k r B m n ( θ , ϕ ) d 3 k ,
V r [ 0 M m n ( p , r ) p 2 d p ] e i k r d 3 r = 2 π 2 i n ( 1 ) m ( n + m ) ! ( n m ) ! B m n ( θ , ϕ ) .
N m n ( k , r ) = 1 k × M m n ( k , r ) = i n + 1 4 π 4 π e i k r C m n ( θ , ϕ ) d Ω .
V r [ 0 N m n ( p , r ) p 2 d p ] e i k r d v = 2 π 2 i n 1 ( 1 ) m ( n + m ) ! ( n m ) ! C m n ( θ , ϕ ) ,
P ˜ n m ( z ) = ( 1 ) m P n m ( z ) .
A n , m = Q 0 ( n , m ) [ k z k Q 1 ( n , m ) k x k Q 2 ( n , m ) ] , B n , m = Q 0 ( n , m ) Q 3 ( n , m ) ,
Q 0 ( n , m ) = [ ( 2 n + 1 ) 4 π ( n m ) ! ( n + m ) ! ] 1 / 2 ( b / R ) 2 n ( n + 1 ) ψ n ( k b ) ,
Q 1 ( n , m ) = 2 π 0 π / 2 sin 2 θ { cos ( k z b cos θ ) i sin ( k z b cos θ ) } P ˜ n m ( cos θ ) × [ I | m 1 | ( i k x b sin θ ) + I | m + 1 | ( i k x b sin θ ) ] d θ ,
Q 2 ( n , m ) = 4 π 0 π / 2 sin θ cos θ { i sin ( k z b cos θ ) cos ( k z b cos θ ) } P ˜ n m ( cos θ ) I | m | ( i k x b sin θ ) d θ ,
Q 3 ( n , m ) = 4 π m k x b 0 π / 2 sin θ { cos ( k z b cos θ ) i sin ( k z b cos θ ) } P ˜ n m ( cos θ ) I | m | ( i k x b sin θ ) d θ ,
T 0 = 0 π J m ( R sin θ sin α ) e ( i R cos θ cos α ) P ˜ n m ( cos θ ) sin θ d θ .
T 0 = 2 i n m j n ( R ) P ˜ n m ( cos α ) ,
J n ( z ) = ( 1 ) n J n ( z ) , j n ( z ) = ( 1 ) n j n ( z ) .
f ( θ ) = sin θ { cos ( k z b cos θ ) i sin ( k z b cos θ ) } P ˜ n m ( cos θ ) I | m | ( i k x b sin θ ) ,
0 π / 2 f ( θ ) d θ = 1 2 0 π f ( θ ) d θ ,
Q 3 ( n , m ) = i n 4 π m sin α j n ( k b ) k b P ˜ n m ( cos α ) .
Q 1 ( n , m ) = ( n + m ) ( n + m 1 ) 2 ( 2 n + 1 ) ( m 1 ) k x b Q 3 ( n 1 , m 1 ) k x b Q 3 ( n 1 , m + 1 ) 2 ( 2 n + 1 ) ( m + 1 ) ( n m + 1 ) ( n m + 2 ) 2 ( 2 n + 1 ) ( m 1 ) k x b Q 3 ( n + 1 , m 1 ) + k x b Q 3 ( n + 1 , m + 1 ) 2 ( 2 n + 1 ) ( m + 1 ) .
Q 2 ( n , m ) = ( n + m ) ( 2 n + 1 ) k x b m Q 3 ( n 1 , m ) ( n m + 1 ) ( 2 n + 1 ) k x b m Q 3 ( n + 1 , m ) .
Q 1 ( n , m ) = T n 2 { j n 1 ( k b ) [ P ˜ n 1 m + 1 ( cos α ) ( n + m ) ( n + m 1 ) P ˜ n 1 m + 1 ( cos α ) ] + j n + 1 ( k b ) [ P ˜ n + 1 m + 1 ( cos α ) ( n m + 1 ) ( n m + 2 ) P ˜ n + 1 m 1 ( cos α ) ] } ,
Q 2 ( n , m ) = T n [ ( n + m ) j n 1 ( k b ) P ˜ n 1 m ( cos α ) ( n m + 1 ) j n + 1 ( k b ) P ˜ n + 1 m ( cos α ) ] ,
Q 1 ( n , m ) = T n sin α j n 1 ( k b ) ( 2 n 1 ) [ ( m 2 + n 2 n ) P ˜ n m ( cos α ) ( n + m ) ( n + m 1 ) P ˜ n 2 m ( cos α ) ] + T n sin α j n + 1 ( k b ) ( 2 n + 3 ) [ ( n m + 1 ) ( n m + 2 ) P ˜ n + 2 m ( cos α ) ( m 2 + n 2 + 3 n + 2 ) P ˜ n m ( cos α ) ] .
Q 1 ( n , m ) = T n sin α cos α [ ( n + m ) j n 1 ( k b ) P ˜ n 1 m ( cos α ) ( n m + 1 ) j n + 1 ( k b ) P ˜ n + 1 m ( cos α ) ] + T n cos α sin α j n ( k b ) k b [ n ( n m + 1 ) P ˜ n + 1 m ( cos α ) ( n + 1 ) ( n + m ) P ˜ n 1 m ( cos α ) ] .
A n , m = [ 1 4 π ( 2 n + 1 ) ( n m ) ! ( n + m ) ! ] 1 2 4 π i n 1 n ( n + 1 ) k 2 R 2 sin α × [ n ( n m + 1 ) P ˜ n + 1 m ( cos α ) ( n + 1 ) ( n + m ) P ˜ n 1 m ( cos α ) ] ,
B n , m = [ ( 2 n + 1 ) 4 π ( n m ) ! ( n + m ) ] 1 / 2 4 π i n m n ( n + 1 ) k 2 R 2 sin α P ˜ n m ( cos α ) .
p m n = ( 1 ) m i n + 1 [ 1 4 π ( 2 n + 1 ) ( n + m ) ! ( n m ) ] 1 / 2 k 2 R 2 A n , m , q m n = ( 1 ) m i n + 2 [ 1 4 π ( 2 n + 1 ) ( n + m ) ! ( n m ) ! ] 1 / 2 k 2 R 2 B n , m .
p m n p = 1 n ( n + 1 ) × [ ( 1 ) m + 1 1 z 2 d d z P ˜ n m ( z ) | z cos α ] = 1 n ( n + 1 ) τ m n ( cos α ) ,
q m n p = 1 n ( n + 1 ) × [ ( 1 ) m m sin α P ˜ n m ( cos α ) ] = 1 n ( n + 1 ) π m n ( cos α ) .

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