Abstract

We derive an analytical expression of the optical torque (OT) on chiral particles in generic monochromatic optical fields within the dipole approximation. Besides the extinction terms owing to the interaction between the dipoles excited on the particle and the incident field, our expression includes also some long missing terms that are understood as recoil due to the radiation of the dipoles excited on the particle. The recoil terms are shown to have a significant contribution to the OT in many situations. Inspired by our expression, we further proved that the net OT vanishes for a lossless isotropic chiral spherical particle of any size illuminated by arbitrary monochromatic optical fields. Finally, we trace the origin of OT on a small chiral sphere immersed in a zeroth-order vector Bessel beam, taking advantage of our analytical expression. It is found that the azimuthal OT perpendicular to the beam’s propagation direction comes from the transfer of the spin angular momentum of the incident field to the particle, while the longitudinal OT along the illumination direction originates from the particle chirality, which generates longitudinal angular momentum on the optical beam and thus makes the particle subject to a longitudinal OT by recoil. Thus the longitudinal OT tends to rotate the chiral particle with opposite helicities in opposite directions, while the transverse OT shows little dependence on the handedness of particle chirality. Our results may help in understanding OT as an extra handle for particle manipulations in optical tweezers.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2017 (3)

A. Rahimzadegan, R. Alaee, I. Fernandez-Corbaton, and C. Rockstuhl, “Fundamental limits of optical force and torque,” Phys. Rev. B 95, 035106 (2017).
[Crossref]

T. H. Zhang, M. Mahdy, Y. M. Liu, J. H. Teng, C. T. Lim, Z. Wang, and C. W. Qiu, “All-optical chirality-sensitive sorting via reversible lateral forces in interference fields,” ACS Nano 11, 4292–4300 (2017).
[Crossref] [PubMed]

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

2016 (1)

F. G. Mitri, “Negative optical spin torque wrench of a non-diffracting non-paraxial fractional Bessel vortex beam,” J. Quantum Spectr. Rad. Trans. 182, 172–179 (2016).
[Crossref]

2015 (9)

M. Nieto-Vesperinas, “Optical torque on small bi-isotropic particles,” Opt. Lett. 40, 3021–3024 (2015).
[Crossref] [PubMed]

A. Canaguier-Durand and C. Genet, “Chiral route to pulling optical forces and left-handed optical torques,” Phys. Rev. A 92, 043823 (2015).
[Crossref]

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

F. Cardano and L. Marrucci, “Spin-orbit photonics,” Nat. Photonics 9, 776–778 (2015).
[Crossref]

A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9, 789–795 (2015).
[Crossref]

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Physics Reports 592, 1–38 (2015).
[Crossref]

A.Y. Bekshaev, K.Y. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5, 011039 (2015).

M. Nieto-Vesperinas, “Optical torque: Electromagnetic spin and orbital-angular-momentum conservation laws and their significance,” Phys. Rev. A 92, 043843 (2015).
[Crossref]

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

2014 (10)

G. Tkachenko and E. Brasselet, “Optofluidic sorting of material chirality by chiral light,” Nat. Commun. 5, 3577 (2014).
[Crossref] [PubMed]

G. Tkachenko and E. Brasselet, “Helicity-dependent three-dimensional optical trapping of chiral microparticles,” Nat. Commun. 5, 4491 (2014).
[Crossref] [PubMed]

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

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

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

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

K. Ding, J. Ng, L. Zhou, and C. T. Chan, “Realization of optical pulling forces using chirality,” Phys. Rev. A 89, 063825 (2014).
[Crossref]

D. Hakobyan and E. Brasselet, “Left-handed optical radiation torque,” Nat. Photonics 8, 610 (2014).
[Crossref]

J. Chen, J. Ng, K. Ding, K. H. Fung, Z. F. Lin, and C. T. Chan, “Negative optical torque,” Sci. Rep. 4, 6386 (2014).
[Crossref] [PubMed]

M. M. Li, S. H. Yan, B. L. Yao, M. Lei, Y. L. Yang, J. W. Min, and D. Dan, “Intrinsic optical torque of cylindrical vector beams on Rayleigh absorptive spherical particles,” J. Opt. Soc. Am. A 31, 1710–1715 (2014).
[Crossref]

2013 (3)

Q. C. Shang, Z. S. Wu, T. Qu, Z. J. Li, L. Bai, and L. Gong, “Analysis of the radiation force and torque exerted on a chiral sphere by a Gaussian beam,” Opt. Express 21, 8677–8688 (2013).
[Crossref] [PubMed]

A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Mechanical separation of chiral dipoles by chiral light,” New J. Phys. 15, 123037 (2013).
[Crossref]

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonics wheel-demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. 8, 13032 (2013).
[Crossref]

2011 (3)

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 12, 053001 (2011).
[Crossref]

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

G. Cipparrone, A. Mazzulla, A. Pane, R. J. Hernandez, and R. Bartolino, “Chiral self-assembled solid microspheres a novel multifunctional microphotonic device,” Adv. Mater. 23, 5773 (2011).
[Crossref] [PubMed]

2010 (1)

J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Progr. Opt. 54, 1–88 (2010).
[Crossref]

2009 (3)

P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17, 2224–2234 (2009).
[Crossref] [PubMed]

M.V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009).
[Crossref]

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103, 100401 (2009).
[Crossref] [PubMed]

2008 (1)

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[Crossref]

2007 (2)

P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical trorque: Application to a micropropeller,” J. Appl. Phys. 101, 023106 (2007).
[Crossref]

A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. A 24, 2844–2849 (2007).
[Crossref]

2005 (1)

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

2004 (1)

M. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Phys. Today 57, 35–40 (2004).
[Crossref]

2003 (1)

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

1998 (2)

M. E. J. Friese, T. A. Niemenen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (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, 286–296 (1998).
[Crossref]

1995 (1)

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[Crossref]

1984 (1)

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

1966 (1)

P. Allen, “A radiation torque experiment,” Am. J. Phys. 74, 1185 (1966).
[Crossref]

1957 (1)

G. T. Di Francia, “On a macroscopic measurement of the spin of electromagnetic radiation,” Il Nuovo Cimento 6, 150–167 (1957).
[Crossref]

1936 (1)

R. E. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[Crossref]

1889 (1)

S. P. Thompson, “Optical torque,” Nature 40, 232–235 (1889).
[Crossref]

Aiello, A.

A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9, 789–795 (2015).
[Crossref]

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonics wheel-demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. 8, 13032 (2013).
[Crossref]

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103, 100401 (2009).
[Crossref] [PubMed]

Alaee, R.

A. Rahimzadegan, R. Alaee, I. Fernandez-Corbaton, and C. Rockstuhl, “Fundamental limits of optical force and torque,” Phys. Rev. B 95, 035106 (2017).
[Crossref]

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[Crossref]

M. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Phys. Today 57, 35–40 (2004).
[Crossref]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP Publishing, 2003).
[Crossref]

Allen, P.

P. Allen, “A radiation torque experiment,” Am. J. Phys. 74, 1185 (1966).
[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, Optical Trapping and Manipulation of Neutral Particles Using Lasers (World Scientific, 2006).
[Crossref]

Bai, L.

Banzer, P.

A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9, 789–795 (2015).
[Crossref]

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonics wheel-demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. 8, 13032 (2013).
[Crossref]

Barnett, S. M.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP Publishing, 2003).
[Crossref]

Bartolino, R.

G. Cipparrone, A. Mazzulla, A. Pane, R. J. Hernandez, and R. Bartolino, “Chiral self-assembled solid microspheres a novel multifunctional microphotonic device,” Adv. Mater. 23, 5773 (2011).
[Crossref] [PubMed]

Bauer, T.

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonics wheel-demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. 8, 13032 (2013).
[Crossref]

Bekshaev, A.

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 12, 053001 (2011).
[Crossref]

Bekshaev, A.Y.

A.Y. Bekshaev, K.Y. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5, 011039 (2015).

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

Berry, M.V.

M.V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009).
[Crossref]

Beth, R. E.

R. E. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[Crossref]

Billaudeau, C.

P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical trorque: Application to a micropropeller,” J. Appl. Phys. 101, 023106 (2007).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Physics Reports 592, 1–38 (2015).
[Crossref]

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 12, 053001 (2011).
[Crossref]

Bliokh, K.Y.

A.Y. Bekshaev, K.Y. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5, 011039 (2015).

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

Bohren, C. F.

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

Bouchal, Z.

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[Crossref]

Brasselet, E.

D. Hakobyan and E. Brasselet, “Left-handed optical radiation torque,” Nat. Photonics 8, 610 (2014).
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G. Tkachenko and E. Brasselet, “Optofluidic sorting of material chirality by chiral light,” Nat. Commun. 5, 3577 (2014).
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K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
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Figures (4)

Fig. 1
Fig. 1 (a) The intensity distribution of the electric field on a transverse plane for a TM-polarized zeroth-order vector Bessel beam propagating along z direction. (b) The OT acting on a chiral nanoparticle with/without loss illuminated by the beam in vacuum versus the particle displacement r from the beam axis. The dashed and solid lines denote the particle with ϵ = 2.5 and ϵ = 2.5 + 0.2i, respectively. The particle has the radius rs = 100 nm, the chirality parameter κ = 0.5, and the permeability µ = 1 for the above cases. The incident wavelength is λ = 1064 nm and the cone angle is α = 30°.
Fig. 2
Fig. 2 The contributions of decomposed azimuthal OT Tϕ1, Tϕ2, and Tϕ3 (a) and decomposed longitudinal OT Tz1, Tz2, and Tz3 (b), defined, respectively, in Eqs. (14) and (15), versus particle displacement r from the beam axis. The accurate total OT FFWS calculated with the FWS approach is plotted for comparison. All other parameters are the same as that of Fig. 1.
Fig. 3
Fig. 3 (a) Time-averaged SAM densities for the incident optical field. (b) Time-averaged Poynting vectors of the total field for a chiral particle with κ = 0.5 located at the coordinate (600,0,0) with ϕ = 0 and immersed in the incident field. The lengths and colors of the arrows denote the amplitude of the SAM density and the energy flux. All other parameters are the same as in Fig. 2.
Fig. 4
Fig. 4 The azimuthal OT Tϕ (a) and the longitudinal OT Tz (b) as the functions of the particle displacement r and the chirality parameter κ. The OT is in the unit of (pN nm) and the white lines denote the zero-value OT.

Equations (29)

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D = ϵ 0 ϵ E + i κ ϵ 0 μ 0 H , B = μ 0 μ H i κ ϵ 0 μ 0 E ,
a n = [ A n ( 2 ) V n ( 1 ) + A n ( 1 ) V n ( 2 ) ] Q n , b n = [ B n ( 1 ) W n ( 2 ) + B n ( 2 ) W n ( 1 ) ] Q n , c n = [ A n ( 1 ) W n ( 2 ) A n ( 2 ) W n ( 1 ) ] Q n ,
A n ( j ) = Z s D n ( 1 ) ( x j ) D n ( 1 ) ( x 0 ) , B n ( j ) = D n ( 1 ) ( x j ) Z s D n ( 1 ) ( x 0 ) , W n ( j ) = Z s D n ( 1 ) ( x j ) D n ( 3 ) ( x 0 ) , V n ( j ) = D n ( 1 ) ( x j ) Z s D n ( 3 ) ( x 0 ) , Q n = ψ n ( x 0 ) / ξ n ( x 0 ) V n ( 1 ) W n ( 2 ) + V n ( 2 ) W n ( 1 ) .
p = α e e E + α e m B , m = α e m E + α m m B ,
α e e = i 6 π ϵ 0 k 0 3 a 1 , α m m = i 6 π μ 0 k 0 3 b 1 , α e m = 6 π Z 0 k 0 3 c 1 ,
T = 1 2 Re ( p × E * ) + 1 2 Re ( m × B * ) + k 0 3 12 π ϵ 0 Im ( p × p * ) + μ 0 k 0 3 12 π Im ( m × m * ) .
1 2 Re ( p × E * ) = 1 2 [ Im ( α e e ) Im ( E * × E ) Re ( α e m ) Re ( E × B * ) Im ( α e m ) Im ( E × B * ) ] , k 0 3 12 π ϵ 0 Im ( p × p * ) = k 0 3 12 π ϵ 0 [ α e e α e e * Im ( E * × E ) + 2 Re ( α e e α e m * ) Im ( E × B * ) + 2 Im ( α e e α e m * ) Re ( E × B * ) α e e α e m * Im ( B * × B ) ] , 1 2 Re ( m × B * ) = 1 2 [ Re ( α e m ) Re ( E × B * ) + Im ( α e m ) Im ( E × B * ) + Im ( α m m ) Im ( B * × B ) ] , μ 0 k 0 3 12 π Im ( m × m * ) = μ 0 k 0 3 12 π [ α e m α e m * Im ( E * × E ) 2 Re ( α m m α e m * ) Im ( E × B * ) + 2 Im ( α m m α e m * ) Re ( E × B * ) α m m α m m * Im ( B * × B ) ] .
T = T rad + T flow + T spin e + T spin m ,
T rad = [ 2 μ 0 Re ( α e m ) + μ 0 k 0 3 3 π ϵ 0 Im ( α e e α e m * ) + μ 0 2 k 0 3 3 π Im ( α m m α e m * ) ] S , T flow = [ μ 0 k 0 3 6 π ϵ 0 Re ( α e e α e m * ) μ 0 2 k 0 3 6 π Re ( α m m α e m * ) ] Im ( E × H * ) , T spin e = [ 2 ω ϵ 0 Im ( α e e ) ω k 0 3 3 π ϵ 0 2 α e e α e m * ω μ 0 k 0 3 3 π ϵ 0 α e m α e m * ] L s e , T spin m = [ 2 ω μ 0 Im ( α m m ) ω μ 0 2 k 0 3 3 π α m m α m m * ω μ 0 k 0 3 3 π ϵ 0 α e m α e m * ] L s m ,
T = 12 π k 0 2 ω Re ( c 1 a 1 * c 1 b 1 * c 1 ) S + 6 π k 0 2 ω Im ( b 1 * c 1 a 1 * c 1 ) Im ( E × H * ) + 12 π ω k 0 3 [ | a 1 | 2 + | c 1 | 2 Re ( a 1 ) ] L s e + 12 π ω k 0 3 [ | b 1 | 2 + | c 1 | 2 Re ( b 1 ) ] L s m .
Re ( c 1 a 1 * c 1 b 1 * c 1 ) = 0 , Im ( b 1 * c 1 a 1 * c 1 ) = 0 , | a 1 | 2 + | c 1 | 2 Re ( a 1 ) = 0 , | b 1 | 2 + | c 1 | 2 Re ( b 1 ) = 0 .
E i n c ( r , ϕ , z ) = E 0 [ i k z k r J 1 ( k r r ) e r + J 0 ( k r r ) e z ] e i k z z ,
S = E 0 2 k 0 k z J 1 2 ( k r r ) 2 Z 0 k r 2 e z , L s e = E 0 2 ϵ 0 k z J 0 ( k r r ) J 1 ( k r r ) 2 ω k r e ϕ , Im ( E × H * ) = E 0 2 k 0 J 0 ( k r r ) J 1 ( k r r ) k r Z 0 e r , L s m = 0 .
T = T rad + T flow + T spin e .
T r = T r 1 + T r 2 ,
T r 1 e r = μ 0 k 0 3 6 π ϵ 0 Re ( α e e α e m * ) Im ( E × H * ) , T r 2 e r = μ 0   k 0 3 6 π Re ( α e e α e m * ) Im ( E × H * ) .
T ϕ = T ϕ 1 + T ϕ 2 + T ϕ 3 ,
T ϕ 1 e ϕ = 2 ω ϵ 0 Im ( α e e ) L s e , T ϕ 2 e ϕ = ω k 0 3 3 π ϵ 0 2 α e e α e e * L s e , T ϕ 3 e ϕ = ω μ 0 k 0 3 3 π ϵ 0 α e m α e m * L s e .
T z = T z 1 + T z 2 + T z 3 ,
T z 1 e z = 2 μ 0 Re ( α e m ) S , T z 2 e z = μ 0 k 0 3 3 π ϵ 0 Im ( α e e α e m * ) S , T z 3 e z = μ 0 2 k 0 3 3 π Im ( α m m α e m * ) S .
T = s n K d σ = s n [ r × T ] d σ = s r × [ T n ] d σ ,
T x = Re [ T 1 ] , T y = Im [ T 1 ] , T z = Re [ T 2 ] ,
T 1 = 2 π ϵ 0 E 0 2 k 0 3 n , m [ ( n m ) ( n + m + 1 ) n 2 ( n + 1 ) 2 ] 1 / 2 t 1 , T 2 = 2 π ϵ 0 E 0 2 k 0 3 n , m m t 2 ,
t 1 = a m n a m + 1 n * + b m n b m + 1 n * 1 2 ( a m n p m + 1 n * + p m n a m + 1 n * + b m n q m + 1 n * + q m n b m + 1 n * ) , t 2 = a m n a m n * + b m n b m n * 1 2 ( a m n p m n * + p m n a m n * + b m n q m n * + q m n b m n * ) ,
a m n = a n p m n + c n q m n , b m n = b n q m n + c n p m n .
T r = T x cos ϕ + T y sin ϕ , T ϕ = T y cos ϕ T x sin ϕ , T z = T z .
T z = 2 π ϵ 0 E 0 2 k 0 3 n , m m t 2   ,
t 2   = [ | a n | 2 + | c n | 2 Re ( a n ) ] | p n | 2 + [ | b n | 2 + | c n | 2 Re ( b n ) ] | q n | 2 + 2 Re [ ( a n c n * + b n * c n c n ) p n q n * ] .
| a n | 2 + | c n | 2 Re ( a n ) = 0 , | b n | 2 + | c n | 2 Re ( b n ) = 0 , a n c n * + b n * c n 1 2 ( c n + c n * ) = 0 .

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